Question

AP Test: Waves: Why is this answer correct?

Initial question:

A transverse periodic wave travels along a string that is stretched along the x-axis. The figure above (not necessary to replicate) shows the y-position of a point on the string as a function of time. What further information, if any, is needed to determine the wavelength of the wave?

Answer I chose:

(D) No further info is needed

Correct answer:

(A) The velocity of the wave.

Why do I need the velocity to determine wavelength?

Answer 1

The y-axis is presumably labeled in distance -- from that you can determine the amplitude of the wave. That is not wavelength.

Explanation:

You could print that figure and measure the length of the wave in mm. But the x-axis of the plot is time, not distance. The y-axis is distance, but it is the distance that the string moves, perpendicular to the length of the string, as the wave passes along the string.

Amplitude is the maximum distance from the medium, or rest, position that a point of the string moves perpendicular to the length of the string as the wave passes through. If you prefer, amplitude is also 1/2 of the distance (perpendicular to the length of the string) between the peak and the valley. Thinking of water waves, amplitude is 1/2 of the height difference between peaks and valleys.

If you measure along the x-axis from any point of your choosing to the next occasion that the identical slope and amplitude exists, that is the time that is called the period, `T`. This is time for a wave cycle to repeat, not a length distance. From period, you can calculate frequency, `f`. So the plot gives you amplitude and period and gives the ability to calculate frequency. But there is no way to calculate the wavelength with only that information.

To find the wavelength, `lambda`, you need the velocity to calculate the distance that the wave travels along the string during T, the time we call period. The formula is

`v = lambda/T`

Notice that the dimensions of `lambda/T` are correct for velocity: distance/time.

Additional info: since frequency is calculated `f = 1/T`, an alternative formula for velocity is

`v = lambda*f`

I hope this helps,
Steve