Question 1

A sinusoidal wave is described by $$y(x,t) = A \cos(kx - \omega t). If $$the wave travels in the +x direction, what is the correct expression for the wave number $$k in $$terms of wavelength $$\lambda$$?

Accepted Answer

$$k = 2\pi / \lambda$$

Question 2

A string is fixed at both ends and has length $$L. $$What is the wavelength $$\$$lambda_1$$ of the fundamental mode of vibration?

Accepted Answer

$$\$$lambda_1$$ = 2L$$

Question 3

The speed of a transverse wave on a string is given by $$v = \sqrt{F/\mu}. If $$the tension $$F in $$the string is quadrupled, what happens to the wave speed?

Accepted Answer

It doubles

Question 4

A pulse traveling on a string reflects from a fixed end. What happens to the pulse upon reflection?

Accepted Answer

It inverts upon reflection

Question 5

Two wave pulses of equal amplitude but opposite direction overlap on a string. What is the result at the point of overlap?

Accepted Answer

Destructive interference

Question 6

For a standing wave on a string fixed at both ends, which of the following is true about the nodes?

Accepted Answer

Nodes are at points where the string never moves

Question 7

A violin string of length $$L = 0.33 m $$vibrates at its fundamental frequency with a wavelength $$\lambda = 0.66 m. $$What is the speed $$v of $$the wave if the frequency is $$f = 659 Hz$$?

Accepted Answer

$$v = 435 m/s$$

Question 8

The principle of superposition states that:

Accepted Answer

The total displacement is the sum of the displacements of individual waves

Question 9

For a string fixed at both ends, the frequency of the $$nth $$harmonic is given by $$f_n = n v / 2L. If$$ $$v is $$the speed of the wave and $$L is $$the length, what is the frequency of the second harmonic?

Accepted Answer

$$f_2 = 2v / 2L$

Question 10

A standing wave is described by $$y(x,t) = (A \sin kx) \sin \omega t. $$What is the condition for nodes along the string?

Accepted Answer

$$\sin kx = 0$$

Question 11

A string of length $$L is $$fixed at both ends. Which of the following wavelengths is NOT allowed for standing waves on the string?

Accepted Answer

$$\lambda = L$$

Question 12

If the mass per unit length $$\mu of a $$string is doubled, what happens to the speed $$v of a $$transverse wave on the string, assuming tension $$F is $$constant?

Accepted Answer

It decreases by a factor of $$\sqrt{2}$$

Question 13

A pulse traveling on a string reflects from a free end. What happens to the pulse upon reflection?

Accepted Answer

It reflects without inverting

Question 14

A string is plucked and produces a standing wave with three antinodes. What is the harmonic number of this mode?

Accepted Answer

n = 3

Question 15

The fundamental frequency of a string fixed at both ends is given by $$f_1$$ = \frac{1}{2L} \sqrt{F/\mu}$$. If the length L$ is halved, what happens to the fundamental frequency?

Accepted Answer

It doubles

Question 16

A string fixed at both ends is vibrating at its third harmonic. How many nodes (including the ends) are present?

Accepted Answer

4

Question 17

A standing wave is established on a string fixed at both ends. If the string is vibrating at its fundamental frequency, how is the length of the string $$L$$ related to the wavelength $$\lambda$$?

Accepted Answer

$$L = \lambda / 2$$

Question 18

A guitar string lies in the x-direction at equilibrium. A sinusoidal wave with amplitude $$A = 0.75 mm $$and frequency $$f = 440 Hz $$travels in the –x direction at $$v = 143 m/s. $$What is the wavelength $$\lambda of $$the wave?

Accepted Answer

$$\lambda = 0.325 m$$

Question 19

Which of the following best describes the relationship between the speed of a mechanical wave and the properties of the medium?

Accepted Answer

The speed is constant and determined by the mechanical properties of the medium

Mechanical Waves and Standing Waves on Strings

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