Question 1

A wave pulse is sent down a taut string as shown in the first image. Which statement best describes the motion of the particles of the string as the pulse passes?

Accepted Answer

The particles move perpendicular to the direction of the wave's travel and return to their equilibrium positions after the pulse passes.

Question 2

For a sinusoidal wave on a string, the displacement at position $$x$$ and time $$t is $$given by $$y(x, t) = y_m \sin(kx - \omega t). $$What is the expression for the transverse velocity $$u of a $$string element at position $$x$$ and time $$t$$?

Accepted Answer

$$u = -\omega y_m \cos(kx - \omega t)$$

Question 3

A wave on a string is described by $$y(x, t) = 0.00327 \sin(72.1x - 2.72t). $$What is the speed of this wave?

Accepted Answer

$$v = 0.0377\ \mathrm{m/s}$$

Question 4

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

Accepted Answer

It doubles.

Question 5

If the frequency of a wave on a string is doubled while the tension and mass per unit length remain constant, what happens to the wavelength?

Accepted Answer

It is halved.

Question 6

The average power transmitted by a sinusoidal wave on a string is given by $$P_{avg} = \frac{1}{2} \mu v \$$omega^2$$ y_m^2$. If the amplitude y_m$ is doubled, by what factor does the average power increase?

Accepted Answer

It increases by a factor of 4.

Question 7

Two waves of equal amplitude $$y_m$$ and frequency travel in the same direction along a string, but with a phase difference $$\phi. $$What is the amplitude of the resultant wave?

Accepted Answer

$$2y_m \cos(\phi/2)$$

Question 8

What phase difference $$\phi$$ between two waves of equal amplitude will result in complete destructive interference?

Accepted Answer

$$\phi = (2n+1)\pi$$

Question 9

A standing wave is formed on a string fixed at both ends. What is the distance between two adjacent nodes?

Accepted Answer

$$\lambda/4$$

Question 10

For a string of length $$L$$ fixed at both ends, what is the wavelength of the fundamental mode (first harmonic)?

Accepted Answer

$$\lambda = 2L$$

Question 11

A string of length $$L$$ fixed at both ends supports a standing wave at its third harmonic. What is the frequency of this mode if the wave speed is $$v$$?

Accepted Answer

$$f = \frac{3v}{2L}$$

Question 12

If a wave travels from a string with mass per unit length $$\mu_1$ to another string with mass per unit length $$\$$mu_2 ($$with the same tension), what happens to the wave speed?

Accepted Answer

It increases if $$\mu_2$ \u003c \mu_1$.

Question 13

A wave on a string is described by $$y(x, t) = y_m \sin(kx + \omega t). In $$which direction is the wave traveling?

Accepted Answer

In the -x direction.

Question 14

Which of the following best describes the energy transport in a wave on a string?

Accepted Answer

Energy is transported along the string by the wave, but the particles of the string only oscillate about their equilibrium positions.

Question 15

If the length of a string fixed at both ends is halved, what happens to the fundamental frequency of the standing wave?

Accepted Answer

It doubles.

Question 16

A string of length $$L is $$fixed at both ends and vibrates in its second harmonic. How many nodes (including the ends) are present?

Accepted Answer

3

Question 17

Which of the following is true about the frequency of a wave as it passes from one medium to another?

Accepted Answer

The frequency remains the same.

Question 18

If two waves of the same frequency and amplitude travel in opposite directions on a string, what type of wave is formed?

Accepted Answer

A standing wave.

Question 19

The phase difference $$\phi$$ between two waves traveling the same direction but separated by a distance $$\Delta x is $$given by:

Accepted Answer

$$\phi = k \Delta x$$

Waves: Properties, Superposition, and Standing Waves

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