This is PHYS 1240 - Sound and Music

Lecture 8

Professor Patricia Rankin

Cell Phones silent

Clickers on

Physics 1240 Lecture 8

Today: Standing waves. Waves on strings

Next time: Waves in pipes – Tyler!

physicscourses.colorado.edu/phys1240 (lecture notes, solutions)

Canvas Site: assignments, administration, grades

Homework – HW4 Due Wed February 12th 5pm

Homelabs – Hlab2 Due Monday Feb 10th 5pm

Last lecture - Debrief

A (normal) mode is a motion where every point moves with the same frequency

𝑳 = λ/2 ;

= 2L

A node is a place where a mode has no motion

An anti-node is a place where a mode has maximum motion

(count to get # of mode)

𝑳 = λ

𝑳 = 3λ/2 ; = 2L/3

𝑳 = nλ/2 ; = 2L/n

Can we summarize relations as an equation? Yes !

𝑣𝑡 = 𝑓𝑛𝑛=𝑓𝑛2𝐿

𝑛

𝑓𝑛 = 𝑛 ∙𝑣𝑡2𝐿

𝑛 = 1, 2, 3, 4, …

𝑳 = nλ/2 ; = 2L/n

Velocity of Transverse Waves on a String

F=tv

tension (N)

mass/length (kg/m)

Book uses vs

But it is not the speed of sound in air!

Vibrating Strings (handy formulae)

• For the 𝑛th harmonic,

𝐿 = 𝑛λ

2

• Recall: 𝑣 = 𝜆𝑓

⇒ 𝑓𝑛 = 𝑛𝑣𝑡2𝐿

• New formula: 𝑣𝑡 =𝑇

𝑚/𝐿=

𝐹

μ

𝑓𝑛 =𝑛

2𝐿

𝑇

𝑚/𝐿

tension

mass per

unit length

What are the units of μ ?

A. m/s

B. kg

C. N

D. kg/m

E. m/s2

F=tv

CLICKER 8.1

What are the units of μ ?

A. m/s

B. kg

C. N

D. kg/m

E. m/s2

F=tv

CLICKER 8.1 D

What are the units of F?

A. m/s

B. kg

C. N

D. kg/m

E. m/s2

F=tv

CLICKER 8.2

What are the units of F?

A. m/s

B. kg

C. N

D. kg/m

E. m/s2

F=tv

CLICKER 8.2 C

1N = 1 kg m/s2 ; N/(kg/m) = m2/s2

If the tension is increased by a factor of 9

what happens to the speed of waves on a

string?

A. Goes up by a factor of 3

B. Goes up by a factor of 4.5

C. Goes up by a factor of 9

D. Goes up by a factor of 81

E. None of these / I don’t know

F=tv

CLICKER 8.3

If the tension is increased by a factor of 9

what happens to the speed of waves on a

string?

A. Goes up by a factor of 3

B. Goes up by a factor of 4.5

C. Goes up by a factor of 9

D. Goes up by a factor of 81

E. None of these / I don’t know

F=tv

CLICKER 8.3 A

If you increase tension by a factor of 4 …

A) The frequency of the fundamental doubles, all

other harmonics stay the same as they were

B) The frequency of every harmonic doubles

C) None of the frequencies change, the

wavelengths double

D) f1 goes up by 2, f2 by 4, (etc…)

E) Something else?

CLICKER 8.4

If you increase tension by a factor of 4 …

A) The frequency of the fundamental doubles, all

other harmonics stay the same as they were

B) The frequency of every harmonic doubles

C) None of the frequencies change, the

wavelengths double

D) f1 goes up by 2, f2 by 4, (etc…)

E) Something else?

CLICKER 8.4 B

A string vibrates with a fundamental frequency of 220 Hz.

Besides 220 Hz, which of the following are resonant

frequencies you might also observe?

i) 110 Hz

ii) 330 Hz

iii) 440 Hz

A: i only

B: ii only

C: iii only

D: i and ii

E: all three

CLICKER 8.5

A string vibrates with a fundamental frequency of 220 Hz.

Besides 220 Hz, which of the following are resonant

frequencies you might also observe?

i) 110 Hz

ii) 330 Hz

iii) 440 Hz

A: i only

B: ii only

C: iii only

D: i and ii

E: all three

CLICKER 8.5 C

A string on an instrument plays an A (440 Hz)

when plucked. If you lightly touch the string

½ way from one end, and then pluck, you are

mostly likely to hear…

A: Still 440 Hz

B: 220 Hz

C: 880 Hz

D: Something entirely different

CLICKER 8.6

A string on an instrument plays an A (440 Hz)

when plucked. If you lightly touch the string

½ way from one end, and then pluck, you are

mostly likely to hear…

A: Still 440 Hz

B: 220 Hz

C: 880 Hz

D: Something entirely different

CLICKER 8.6C

Octaves

When you increase frequencies by an octave you

double the frequency

Two octaves higher means an increase by a factor

of four

Three octaves higher is a factor of 8 (= 2x2x2 = 23 )

When things increase in this way (according to a

power) we have an example of an exponential

growth

A string has a fundamental frequency f1.

The second mode has frequency f2 = 2f1, or one

octave higher.

Which mode is TWO octaves above f1?

A) f3 B) f4 C) f5

D) f8 E) ??

CLICKER 8.7

A string has a fundamental frequency f1.

The second mode has frequency f2 = 2f1, or one

octave higher.

Which mode is TWO octaves above f1?

A) f3 B) f4 C) f5

D) f8 E) ??

CLICKER 8.7 B