Mechanical Waves Types of waves
One-Dimensional Traveling WavesWave speed in a stretched stringReflection and transmission of wavesEnergy Transmitted by sinusoidal Waves along a string
All waves carry energy, but the amount of energy transmitted through a medium and the mechanism of transport responsible differ from case to case.
Eg. The power of ocean waves during a storm is much greater than the power of sound waves
generated by a single human voice. All mechanical waves require:
(1) some source of disturbance,
(2) a medium that may be disturbed,
(3) some physical mechanism through which elements of the medium can influence each other.
MECHANICAL WAVESo They are governed by Newton’s laws, and they can
exist only within a material medium Eg. Water wave, wave on a string and sound wave
Eg. Sound waves.
LONGITUDINAL WAVESThe motion of the element of the wave (or medium) is parallel to the direction of travel of the wave
Eg. Waves on a string and electromagnetic waves.
TRANSVERSE WAVESThe motion of the element of the wave (or medium) is perpendicular to the direction of travel of the wave
TYPES OF WAVES MOTION:
The motion of water elements on the surface of deep water
in which a wave is propagating is a combination of
transverse and longitudinal displacements, with the result
that elements at the surface move in nearly circular paths.
Each element is displaced both horizontally and vertically
from its equilibrium position.
Q: How is the energy of an earthquake transported?
http://resonanceswavesandfields.blogspot.com/2007/12/types-of-waves-i.html
Amplitude, A : the magnitude of the maximum displacement of the elements from their equilibrium positions as thewave passes through them.
Wavelength, l : the distance between two successive identical points on the wave (eg. distance between two successive crests)
Frequency, f : the number of complete cycles per unit time
Period, T : the time required for 1 complete oscillation; T = 1/f
Wave velocity, v : the velocity at which any part of the wave (eg. the crest) moves; v = fl
Phase of the wave: the argument (kx –wt) of the sine function.
Phase constant f : The value of f chosen so that the function gives some other displacement and slope at x =0 when t = 0.
WAVE CHARACTERISTICS
As if taking a still picture of a wave; amplitude and wavelength may be determined but not frequency and period.
As if observing just a single point on a wave; amplitude and period may be determined but not wavelength .
Representations
The wave equation can also be written as
)sin( tAx
displacementamplitude angular frequency
time
)(cos tAy
Sine wave
)(sin tAy
phase angle
or
Displacement x(t)
0
-A
A
1/4T 1/2T 3/4T T 3/2T
x=A cos(t +)
Am
plit
ude
Period T (second)
Time (second)
The equation of motion is sinusoidal as a function of time
In this case, =/2, x = A cos (t - /2) or x= A sin t
Wave equation (Function of time)
ONE DIMENSIONAL TRAVELING WAVE Pulse traveling to the right
Pulse traveling to the left
The function y, sometimes called
the wave function, depends on
the two variables x and t.
vtxinAtxy2
s),(
T
txinAtxy
λ2s),( )(s),( ωtkxinAtxy
Phase
xAxytAt2
sin)0,( ,s 0
k (= 2 /p l) is angular wave number and w (= 2p/T) is angular frequency
Consider a one dimensional wave traveling along the x-axis:
Suppose the wave move to the right at velocity v. At t s later, the whole wave has moved to the right a distance, vt
The wave equation assumes that the displacement is zero at t = 0. However if the displacement is not zero at t = 0, we generally express the wave in the form:
where f is the phase constant
)(s),( ωtkxinAtxy
Eg. A sinusoidal wave traveling in the positive x direction has an amplitude of 15.0 cm, a wavelength of 40.0 cm, and a frequency of 8.00 Hz. The vertical displacement of the medium at t = 0 and x = 0 is also 15.0 cm, (a) Find the angular wave number k, period T, angular frequency, and speed v of the wave. (b) Determine the phase constant , and write a general expression for the wave function.
Ans; a) k = 2p/ l = 2p rad/ 40.0 cm = 0.157 rad/cm
T =1/f = 1/(8.00 s-1) = 0.125 s
w = 2pf = 2p(8.00 s-1) = 50.3 rad/s
v =f l = (40.0 cm)(8.00 s-1) = 320 cm/s
b) Because A = 15.0 cm and because y = 15.0 cm at x = 0 and t = 0, substitution into the wave function gives
15.0 = (15.0)sin f or sin f = 1
We may take the principal value f = p/2 rad (or 90°). Hence, the wave function is of the form
y = Asin(kx – wt + p/2) = Acos(kx- wt)
Substituting the values for A, k, and into this expression, we obtain
y = (15.0 cm) cos(0.157x - 50.3t )
Example, Transverse Wave
Example, Transverse Wave, Transverse Velocity, and Acceleration
Linear wave equation in general:
Refer Serway page 479
For waves on strings, y represents the vertical position of elements of the string. For sound waves propagating through a gas, y corresponds to longitudinal position of elements of the gas from equilibrium or variations in either the pressure or the density of the gas. In the case of electromagnetic waves, y corresponds to electric or magnetic field components.
Wave Speed on a Stretched StringThe speed of a wave along a stretched ideal string depends only on the tension and linear density of the string and not on the frequency of the wave.
A small string element of length Dl within the pulse is an arc of a circle of radius R and subtending an angle 2q at the center of that circle. A force with a magnitude equal to the tension in the string, t, pulls tangentially on this element at each end. The horizontal components of these forces cancel,but the vertical components add to form a radial restoring force . For small angles,
If m is the linear mass density of the string, and Dm the mass of the small element,
The element has an acceleration:
Therefore,
Eg. A uniform string has a mass of 0.300 kg and a length of 6.00 m. The string passes over a pulley and supports a 2.00-kg object. Find the speed of a pulse traveling along this string.
Reflection and Transmission
(phase change) (no phase change)
For a sinusoidal wave of frequency f, the particles move in a SHM, and each particle has an energy
For a 3-D elastic medium, assuming the entire medium has the uniform density r, Thus m =rV = rA’l = rA’vt; giving
Therefore, energy transported by a wave is proportional to the square of the frequency and to the square of the amplitude.
particle of mass is m where,2
1
m
kfBecause
22'22 AvtfAE
Energy and Power of a Wave
𝑓 =𝜔
2𝜋ThereforeSince
22
2222'
2
1
2
1
2
1
Avt
Avtl
mAvtAE
Power22
2
1Av
t
EP
Energy and Power Transmitted by sinusoidal waves on a string
Example, Transverse Wave:
Example:
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