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AP Physics – Vibrations, Waves and Sound Waves – Chapter 14-15 Chapter 14 - Reading pp. 421-448 - text HW #1, 7, 8, 13, 20, 23, 25 (a,b,d), 35, 36, 39, 41, 42, 43, 45, 48 #1) A 5 kg block slides down a frictionless incline, collides with and sticks to a board fastened to a spring with a constant 20 N/m

(a) Find the maximum amount the spring is compressed. (b) Describe what happens to the block after it sticks to the spring.

The block will oscillate back and forth in SHM – Simple Harmonic Motion. Simple Harmonic Motion A type of motion that results when the net force on a moving object is a Hooke’s law type of force (a RESTORING force that is proportional to the displacement but in the opposite direction The two most frequently studied harmonic oscillators:

1. The Mass-Spring System 2. The Pendulum Demo: Mass-Spring Oscillator on air track

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#2) A spring is initially at the equilibrium position. It is then pulled down by an external force and released. Draw a position vs. time graph that represents this motion.

Pendulum - Simple Harmonic motion applet Important physical quantities describing SHM:

1) Amplitude (A) - Maximum displacement from equilibrium position (Unit: meter)

2) Period (T) – Time for one complete oscillation or cycle. (Unit: sec/cycle)

3) Frequency (f)– the number of cycles per unit of time (Unit: cycles/s, s-1, or hertz (Hz))

4) Wavelength (λ) – The distance between two consecutive points on a wave that are in phase (moving in same direction with same amplitude)

What is the relationship between the frequency of an object and its period?

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An in-depth analysis of the motion of a mass on a spring oscillating horizontally:

From Hooke’s Law and Newton’s Laws: F = ma and F =-kx Hence -kx = ma a = + kx m Notice: acceleration is not constant and varies from –kx to +kx m m How would the position vs. time, velocity vs. time, and acceleration vs. time graphs look? Compare mass spring to circular motion here

Equation of curve: x = A cos(ωt) (recall ω=θ/t or θ= ωt) v = - ωA sin(ωt) a = - ω2A cos(ωt)

(remember: Fnet and a are in same direction)

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#3) The following is a position time graph for an oscillating pendulum:

Find the Amplitude, Period and Frequency. #4) AP Exam 1998 #38

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Potential Energy (U), Kinetic Energy (K) and total Mechanical Energy Graphs for objects in SHM. #5) A block oscillates along a frictionless, horizontal surface

Graph:

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B) Repeat for a vertical spring. (next page) (Reference Point: PEg=0 J at the equilibrium position) Blue – Elastic PE Green – Gravitational PE Red – Kinetic Black -Total Mechanical Now go home and do homework - # 1, 7, 8, 13

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14.4 Period of a Mass-Spring System (Works for both vertical and horizontal) (Class demo lab) Compare mass spring to circular motion here

What are the factors that effect the period for a mass oscillating on a vertical spring? The equation: The equation derived: Since the horizontal projection of an object in circular motion is identical to simple harmonic motion, we can use the equation for linear velocity as a starting point:

or or

Now consider ¼ of the cycle of the mass spring as it moves from equilibrium position (max KE), to the fully stretched position (max PEs)

or or

Since r=x, combining eq (1) and (2) yields:

k

mT 2

T

rv

2

v

rT

2

22

2

1

2

1mvkx

k

m

v

x

2

2

k

m

v

x

v

rT 2

k

m

v

xT 22

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#6) (A) A 2-kg mass oscillates on a mass-spring system with spring constant 18 N/m. What is the period and frequency? B) If the period is 6 s for this spring, what mass was used? #7) (A)When a mass is attached to a spring, the period of oscillation is approximately 2 s. When the mass attached to the spring is doubled, the period of the pendulum is nearly

(A) 0.5 s (B) 1.0 s (C) 1.4 s (D) 2.0 s (E) 2.8 s (B) This new mass is submerged in a liquid, and set into oscillation. The period of the pendulum is nearly (A) 0.5 s (B) 1.0 s (C) 1.4 s (D) 2.0 s (E) 2.8 s Note: This is an example of a damped oscillator. In this case, the period does not change, but it quickly brings the oscillator to rest, which is important in situations where unwanted vibrations occur. Most common dampener you utilize every day? The shock absorbers on your car. When your wheel hits a bump, the springs allow the wheel to move without transferring all the energy to the interior compartment. But, the wheel would bounce up and down in SHM if the car did not have shock absorbers, which are oil filled pistons (similar to the piston on a screen door that keeps it from slamming). The shocks limit the spring to 2-3 oscillations. Now go home and do homework - # 20, 23, 25abd

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14.6 Period of a Pendulum What are the factors that effect the period for a simple pendulum oscillating at small amplitudes? The equation: The equation derived: We start with the restoring force (Ftangent), which remember is:

Ftangent = mgsinθ For small angles, (<15°) sinθ = θ or Ftangent = mgθ

Since θ=s/r Since the Ftangent equation above is of the form Fs = -kx, mg/l is equivalent to k. Thus mg/l can be subbed into the equation for the period of a mass spring system:

or

g

lT 2

l

mg

m

k

mT 22

sl

mgF tan

g

lT 2

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#8) A) Three physics students of combined mass of 200 kg oscillate on the Daredevil Dive at Great Adventure with cable length of 47 m. What is the period and frequency of this pendulum on Earth? B) If we moved this pendulum to the moon (at great monetary expense) gmoon = 1.63m/s2, what is the period? #9) The pendulum of a Grandfather clock requires a period of 2 s to keep accurate time for a pendulum on Earth, what length pendulum is needed? What length would be needed on Jupiter (g= 26 m/s2) #10) A pendulum with mass M and length L on Earth, oscillates with a period T. If the length is doubled, the period is

Now go home and do homework - #35, 36, 39

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14.8-14.9 Wave Motion and types of waves Important definitions: 1)Wave – the periodic motion of a medium due to a vibrating object. A wave always TRANSFERS THE ENERGY of the vibrating object but does not transfer mass. 2)MEDIUM – The substance or material through which the wave travels. Hmmm…Matter medium vs. empty space: is empty space a medium? Let’s scour the Web for theories on this. 3)Transverse Wave – a wave in which the particles of the disturbed medium move perpendicular to the direction of wave velocity.

Examples of Transverse waves: 1) Water waves (actually a combination of transverse

and longitudinal) 2) Electromagnetic Waves (no matter medium

required) Check out Physics applet: (transverse waves) 4)Pulse: A single vibratory disturbance #11 (A) What happens to point p as the pulse passes through it?

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(B) Pulse on a string – fixed vs free-end reflection Fixed end: Inversion occurs (180° change of phase) Free end: no inversion

(Applet pulse on a string – reflected waves)

Which of Newton’s law applies to reflected pulse being inverted? Newton’s third Law of Action-Reaction. The last molecule in the string is attached to the pole, which acts as a really BIG molecule. Thus the molecule in the string pulls up on the pole, while the pole simultaneously pulls down on the molecule in the string. Since the pole is so massive, it’s motion changes negligibly, while the

strings motion changes greatly and is thus pulled DOWN. 5)Longitudinal Wave – a wave in which the particles of the disturbed medium move parallel to the direction of the wave velocity.

Example of Longitudinal waves: 1) SOUND WAVES

DEMO: Use slinky as a Longitudinal Wave (longitudinal wave applet)

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14.10Relationship of Frequency, Wavelength and Speed of a wave Derive the formula for the speed of a wave in terms of frequency and wavelength. Remember, that waves travel at constant speed within the same medium. #12)Find the amplitude, frequency, period and wavelength

Since the length of 1 cycle = λ and the time to complete 1 cycle = T

and v = d/t then:

Tv

which leads to f

v1

or fv

In a uniform medium, the relationship between f and λ is inverse. Demo with wave spring: vary f and watch λ change.

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#13) Find the velocity of the wave in the string below.

Does the answer make sense? Yes if you CHANGE THE MEDIUM!

BIG NOTE:

The VELOCITY of a wave is dependent on the

MEDIUM, ONLY.

In a uniform rope:

tFv

Examples: Will a wave travel faster through a rope stretched tightly or one hanging loosely?

(Tight rope since more tension increases speed) Will a wave travel faster through a thick rope or a thin rope?

(Thin rope - since linear density is inversely related

to speed (Applet pulse on a string – thick to thin)

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Let’s test this equation in the hall: #14 (A) Find the speed of a pulse on a uniform string of mass ____ kg and length ____ m when subjected to a tension of _____ N. (B) How long will it take the pulse to travel the _____ m? If the tension were doubled, the speed would now be_________.

Now go home and do homework - #41,42,43,45,48

When are 2 points on a wave in PHASE?

Name 2 points on the wave that have a 0o phase difference __________ 180o phase difference ____________

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Two points on a wave are in phase when they have the same displacement from the equilibrium position AND are moving in the same direction. In terms of wavelengths: 0o phase difference: nλ apart (n=0, 1, 2,…) 180o phase difference: nλ/2 apart (n=1, 2, 3,…) 14.12Superposition and Interference What happens when 2 or more waves are passing through the same medium? Predict what happens: #15) Draw the resulting wave at the instant both waves are at pt. p. This is an example of CONSTRUCTIVE interference. (Demo: in phase pulse)Applet

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#14) Draw the resulting wave at the instant both waves are at pt. p

This is an example of DESTRUCTIVE interference. (Demo: out-of-phase pulse) Applet The Superposition Principle – When 2 or more wave are traveling through the same medium, they interfere with each other. The resulting waveform is found by ADDING the displacements of the individual waves together, point by point. #15) Draw the resultant wave when the dotted lines align at point p

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#16)This diagram shows two theoretical pulses headed towards each other. The pulses each travel one grid block in one time unit.

Draw the resulting pattern one unit after original diagram.

Draw the resulting pattern two units after original diagram.

Draw the resulting pattern three units after original diagram.

Draw the resulting pattern four units after original diagram.

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15.7 Interference of Sound Waves

#17) Two speakers are in phase and each produce a sound with a frequency of 120 Hz. The diagram below is the representation of sound on an oscilloscope screen.

What is the minimum non zero distance “d” that either speaker can be shifted backwards so that: (Use vsound in air @ STP = 330 m/s) (A) Maximum constructive interference occurs.

(B) Maximum destructive interference occurs.

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AP Physics – Sound Waves – Chapter 15 Chapter 15 - Reading pp. 458-487 - text HW #4, 21, 22, 24, 31, 37, 38, 39, 40, 44, 47, 48, 50, 53 Sound – the most important example of a longitudinal wave. The direction of motion of the medium is PARALLEL to the direction of the wave velocity. Sound waves require a matter medium! Sound waves in air consist of a series of: Compressions or Condensations (high density region of air molecules) and Rarefactions (low density region of air molecules).

(Applet: longitudinal waves)

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For a sound wave: The louder the sound, the greater the wave’s AMPLITUDE. The higher the pitch the greater the wave’s FREQUENCY. Speed of sound in air – remember this equation: Where T is measured in Celcius! #18 A) Calculate the speed of sound in air at STP. B) Calculate the speed of sound in air at room temperature (68o F) or ____oC #19) Middle C on the piano has a frequency of 264 Hz and a wavelength of 1.3 m. Find the speed of sound in air. Determine the temperature of the room that the piano was played.

What is Echo? - A reflected sound wave. #20) Sonar (originally an acronym for SOund Navigation And Ranging) is used to locate underwater obstacles. A sonar ping is sounded from a Naval vessel and is heard by the sonar officer 0.4 sec later. How far away is the obstacle, given the speed of sound in water as 1500 m/s?

2731)/331(

Tsmv

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15.6 The Doppler Effect The Doppler effect is experienced whenever there is relative motion between the source of sound and observer. If relative motion between source and observer is towards each other, the observed/apparent frequency is greater than actual. When the relative motion between the source and observer is away from each other, the observed or apparent frequency is less than actual. Doppler Applet Link

General Formula:

Moving Towards = ___ Moving Away = ___ (Don’t use words like closer or approaches) (Doppler Videos)

Case I: Case I Formula:

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Case II: Case II Formula:

Case III: Case III Formula:

Case IV: Case IV Formula:

Case V: Case V Formula:

Case VI: Case VI Formula:

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#21) A train emits a 400-Hz horn blast. Use the speed of sound in air as 343 m/s. If the train is moving at 36 m/s, what is the frequency of the sound perceived by an observer standing beside the track? (A) as the train approaches. (B) after the train has passes the observer and is moving away. #22) A tea kettle sitting on the stove blows its whistle with a frequency of 500 Hz. Mr. Berbit, a big fan of tea, taps into his old-man strength and races to the stove at 12 m/s to shut it off. (assume room temperature: vsound=343 m/s) A) What frequency does Mr. Berbit hear? B) What is the apparent wavelength of the observed frequency? #23) Responding to a 911 call for massive 3rd degree burns at the old Berbit house, An ambulance moving at 45 m/s emits a siren with a frequency of 400 Hz (at STP). What is the frequency observed by a person in a car traveling at 30 m/s when: (A) Both are moving away from each other? (B) Both are moving towards each other? C) What is the frequency observed by the ambulance driver in case A and B?

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#24) The same ambulance moving at 45 m/s north with Mr. Berbit inside emits a siren with a frequency of 400 Hz. A car ahead of the ambulance is also moving north at 30 m/s. A) What is the frequency observed by the person in a car? (Remember, the car moves away from the ambulance and the ambulance is moving towards the car!) Key: Draw picture….is the object (whether or not it is the source or observer) moving towards or away? Towards causes f ' to increase, away causes f ' to decrease B) If the car behind the ambulance is moving north at 30 m/s, what frequency would be observed by a passenger in the car? C) If the car that was ahead of the ambulance had the same siren, what frequency would a person in the ambulance hear? Notice (A) > 400 Hz,(B) <400 Hz,(C) >400 Hz

Now go home and do Homework - #21, 22, 24, 31

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#24) A small vibrating object moves across the surface of a ripple tank producing the wave fronts as shown. The wave fronts move with speed v. The object is traveling in what direction and with what speed relative to the speed of the wave fronts produced?

Direction Speed A) To the right Equal to v B) To the right Less than v C) To the right Greater than v D) To the left Less than v E) To the left Greater than v Check out Physics Applet for answer: Doppler Applet Link #25) Now draw the wave when A) vsource = 0 B) vsource < vsound and source moves to right C) vsource = vsound and source moves to left D) vsource > vsound and source moves to right Shock wave? (When vsource > vsound) Sonic Boom Clip FYI: Mach Number is the ratio of vsource:vsound

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15.8 What is a Standing Wave (In General)? When waves are confined in a space, like waves on a guitar string or in an organ pipe, there are reflections at both ends, leading to WAVES traveling in OPPOSITE directions with the SAME AMPLITUDE and FREQUENCY. These waves combine according to the superposition principle. For a given length string or pipe, there are specific frequencies for which this interference results in a stationary vibration pattern called a Standing Wave. DEMO: Standing Wave using a slinky Standing Wave on a String) Standing Wave on a String “Continuous” Standing Wave in a guitar string:

1st Harmonic: (the fundamental frequency)

# of nodes _____ L = ____λ1 # of antinodes _____

2nd Harmonic:

# of nodes _____ L = ____λ2 # of antinodes _____

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3rd Harmonic:

# of nodes _____ L = ____λ3 # of antinodes _____

Since velocity of a wave on a string is

and

What is each frequency in terms of L? In terms of f1? f1 = v f1 = λ1 f2 = v f2 = λ2 f3 = v f3 = λ3 Hence, General Formula for nth frequency

#26) Given: f48 = 312 Hz find f5.

tF

v fv

t

n

F

Ln

L

vnf

2

1

2

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#27) A piano string is 1.10 m long and has a mass of 9.0 grams. A) How much tension must the string be under if it is to vibrate at a fundamental frequency of 131 Hz? B) Find f2, f3, f4. C) If the tension is quadrupled, what is the new fundamental frequency f1? f2? f3? Interesting aside: Frequency Matching. Why do Tires on cars sometimes

seem to run backwards when watching them on the highway at night or in a video? That’s because they are either being illuminated by a vapor lamp hooked up to AC current (which switches on and off at 60 Hz) or because they are being filmed by a camera taking pictures at a certain rate. In either case, you are witnessing a stroboscopic effect called frequency matching. Here is a Wiki description: Special calibrated strobe lights, capable of flashing up to hundreds of times per second, are used in industry to "stop" the motion of rotating and other repetitively-operating machinery and to measure, or adjust, the rotation speeds or cycle times. Since this stop is only apparent, a marked point on the rotating body will either appear to move backward or forward, or not move, depending on the frequency of the strobe-flash. If the flash occurs equal to the period of rotation (or evenly multiplied or divided, i.e. n*ω or ω/n, where n is an integer and ω the angular frequency) the marked point will appear to not move. Any non integer flash setting will make the mark appear to move forward or backward, e.g. a slight increase of the flash frequency will make the point appear to move backward. DEMO: Stroboscope and motor

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#28) A frequency generator is hooked up to a string under Tension T due to mass M (___kg) being hung on the string of linear density μ (___kg/m).

(A) For the given tension T, determine the fundamental frequency. B) Predict f2, f3 etc. C) If the tension is increased, the fundamental frequency __________. D) If the length of string is shortened (tension remains the same), the fundamental frequency ___________. Use the Stroboscope on the standing wave

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#29) A cord of fixed length and uniform density, when held between two fixed points under tension T vibrates with a fundamental frequency f. If the tension is doubled the 2nd harmonic is:

#30) A wire fixed at both ends is set into vibration. If the frequency is 60 vibrations per second, the velocity of propagation of the transverse wave along the wire is A) 30 m/s B) 60 m/s C) 120 m/s D) 180 m/s E) 240 m/s Now go home and do homework #37, 38, 39, 40, 44

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15.10 Standing Waves in Air Columns Standing LONGITUDINAL waves can be set up in a tube of air, open at both ends (such as an organ pipe or flute), or closed at one end (such as a saxomophone or glass bottle). This is the result of two waves traveling in

opposite directions, with the same frequency. See website: http://www.phys.unsw.edu.au/jw/flutes.v.clarinets.html (scroll down to see how sound is reflected at the closed end, and even open end) Case I (Pipe Open at Both Ends) If the end of the pipe is open, the pipe vibrates, and thus, the air molecules have complete freedom of motion and a(n) ___________ exists at the end of the pipe. Draw the first 4 Harmonics: 1st Harmonic: (Fundamental Frequency)

# Nodes _____ # Antinodes _____ L = ____λ1

____________________________________________

2nd Harmonic:

# Nodes _____ # Antinodes _____

L = _____ λ2

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3rd Harmonic:

# Nodes _____ # Antinodes _____

L = _____ λ3

THUS, in general, for a pipe open at both ends, the formula is:

for n = 1, 2, 3, … Pipe organ applet:

Case II (Pipe Closed at One End) If the end of the pipe is closed, the pipe vibrates, but the air molecules do NOT have complete freedom of motion and a(n) _________ exists at the closed end of the pipe. Draw the first 4 Harmonics: 1st Harmonic:(Fundamental Frequency)

# Nodes _____ # Antinodes _____

L = ____λ1

L

vnfn

2

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___Harmonic:

# Nodes _____ # Antinodes _____ L = ____λ

____________________________________

___ Harmonic:

# Nodes _____ # Antinodes _____

L = ____λ Thus, for a pipe closed at one end, the general formula is:

for n = 1, 3, 5, … Pipe organ applet:

Demo: Metal and cardboard pipes open at both ends/Bunsen burner

#31) A pipe has a length of 3m (open at both ends). Determine the frequencies of the first three harmonics if the speed of sound in air is 343 m/s. #32) A pipe has a length of 3m (closed at one end). Determine the frequencies of the first three harmonics if the speed of sound in air is 343 m/s.

Now go home and do homework- #47,48,50,53

L

vnfn

4

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#33) 1995 AP Part B #6 A hollow tube of length L is open at both ends as shown and is held in midair. A tuning fork with a frequency fo vibrates at one end of the tube and causes the air in the tube to vibrate at its fundamental frequency. Express your answers in terms of L and fo. A) Determine the wavelength of the sound. B) Determine the speed of sound in the air inside the tube. C) Determine the next higher frequency at which this air column would resonate. #33 is continued on next page.

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The tube is submerged in a large, graduated cylinder filled with water. The tube is slowly raised out of the water and the same tuning fork, vibrating at fo is held a fixed distance from the top of the tube. D) Determine the height h of the tube above the water when the air column resonates for the first time. Express your answer in terms of l. This Brings us to…

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15.9 Resonance Resonance demos: Singing Pipe, Singing wine glass, Chladni Plates, etc.

Resonance is the tendency of a system to oscillate at maximum amplitude at a certain frequency. This frequency is known as the system's natural frequency of vibration or resonant frequency. Note: If two objects have the same natural frequency of vibration, vibrating one object will cause the other to vibrate, even if they are not touching. This vibration will lead to an increase in ___________. (This is called sympathetic vibrations.)

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15.11 What are beats? 2 waves traveling in the same direction with slightly different frequencies. As time passes, waves A and B move in and out of phase with each other periodically, as shown in the diagram below. So what does the observer hear? (Demonstrate with two tuning forks)

A listener at some fixed point will hear an alternation in loudness known as BEATS. The beat frequency equals the difference in frequency between the two sources. #34) Two tuning forks, each with a frequency of 100 Hz and 97 Hz respectively. Calculate the beat frequency. Chapter 14 and 15. Done.