Question 1

A 2.50-m-long string is vibrating in a standing wave pattern with nodes at 0 m, 1.00 m, 1.50 m, and 2.50 m, and no nodes in between these points. If the frequency of this mode is 100 Hz, what is the harmonic number (mode number) of this standing wave?

Accepted Answer

$$m = 5$$

Question 2

Given the same string as above (2.50 m long, 100 Hz for the fifth harmonic), what is the fundamental frequency $$f_1 of $$the string?

Accepted Answer

$$f_1 = 20$$\ \mathrm{Hz}$$

Question 3

What is the speed $$v of $$the traveling waves on the string described above?

Accepted Answer

$$v = 200\ \mathrm{m/s}$$

Question 4

A guitar string is 0.640 m long and has a mass of 2.86 g. If it is tuned to a fundamental frequency of 110 Hz, what is the tension $$T in $$the string? (Use $$\mu = m/L$$ and $$f_1$$ = \frac{1}{2L}\sqrt{T/\mu}$$)

Accepted Answer

$$T = 225\ \mathrm{N}$$

Question 5

A helium-neon laser emits light of wavelength $$\lambda = 632.8\ \mathrm{nm} in a $$cavity of length $$L = 15.0\ \mathrm{cm}. $$What is the mode number $$m of $$the standing wave in the cavity?

Accepted Answer

$$m = 474,000$$

Question 6

Which of the following best describes the boundary conditions for standing sound waves in a tube that is open at both ends?

Accepted Answer

Both ends are nodes of pressure.

Question 7

For a tube open at one end and closed at the other, which harmonics are present in the standing wave pattern?

Accepted Answer

Only odd harmonics (m = 1, 3, 5, ...)

Question 8

The ear canal can be modeled as a tube open at one end and closed at the other, with a length of 2.5 cm and speed of sound $$v = 350\ \mathrm{m/s}. $$What is the fundamental frequency $$f_1 of $$the standing wave in the ear canal?

Accepted Answer

$$f_1 = 1400$$\ \mathrm{Hz}$$

Question 9

Which equation gives the allowed frequencies for standing sound waves in an open-open tube of length $$L$$ and speed of sound $$v$$?

Accepted Answer

$$f_m = m\left(\frac{v}{2L}\right)$$

Question 10

A flute is modeled as an open-open tube. If the speed of sound is $$340\ \mathrm{m/s}$$ and the tube is $$0.68\ \mathrm{m}$$ long, what is the fundamental frequency?

Accepted Answer

$$f_1 = 500$$\ \mathrm{Hz}$$

Question 11

Which of the following statements about interference is correct?

Accepted Answer

Constructive interference occurs when the path-length difference is a whole number of wavelengths.

Question 12

A standing wave on a string has a node spacing of $$0.50\ \mathrm{m}. $$What is the wavelength of the wave?

Accepted Answer

$$\lambda = 1.00\ \mathrm{m}$$

Question 13

Which of the following best describes the principle of superposition?

Accepted Answer

The displacement of a medium is the sum of the displacements due to each wave.

Question 14

A tube closed at one end and open at the other has a length $$L. $$What is the wavelength of the fundamental mode?

Accepted Answer

$$\$$lambda_1$$ = 4L$$

Question 15

A wind instrument player opens a hole halfway down the tube. What happens to the effective length of the tube and the pitch produced?

Accepted Answer

The effective length decreases and the pitch increases.

Question 16

Which of the following is true for the pressure variation at the open end of a tube supporting a standing sound wave?

Accepted Answer

The pressure is always at a minimum (node).

Question 17

If the speed of sound in air is $$343\ \mathrm{m/s}$$, what is the frequency of the third harmonic in an open-closed tube of length $$0.25\ \mathrm{m}$$?

Accepted Answer

$$f_3 = 1029$$\ \mathrm{Hz}$$

Standing Waves, Harmonics, and Resonance in Physics with Calculus

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