After determining the frequency of the waves in the previous problem, you then use a handheld sonar system to determine the ocean depth at $10.0-\mathrm{m}$ increments from the shore. The data are shown in the table. Complete the table and construct a graph of the wave speed as a function of the depth.

Distance $(\mathrm{m})$	Depth $(\mathrm{m})$	Speed $(\mathrm{m} / \mathrm{s})$	Wavelength $(\mathrm{m})$
$10.0$	$1.4$
$20.0$	$2.8$
$30.0$	$4.2$
$40.0$	$5.6$

A speaker produces a tone at a frequency of $660 \mathrm{~Hz}$. The speaker cone has a maximum displacement of $0.3 \mathrm{~cm}$ when turned on. Assume the speed of sound in air is $343 \mathrm{~m} / \mathrm{s}$. Construct mathematical and graphical representations of the resulting wave.

A speaker produces a tone at a frequency of $500 \mathrm{~Hz}$. The speaker cone has a maximum displacement of $0.7 \mathrm{~cm}$ when turned on. Assume the speed of sound in air is $343 \mathrm{~m} / \mathrm{s}$. Construct mathematical and graphical representations of the resulting wave.

The data show the densities for two different ropes and the wavelength, frequency, and speed of a wave traveling on each rope. Based on the data, which of the three other properties appears to cause a change in the speed?

What properties could you use to define an ocean wave? Specifically, if you had to describe waves coming on shore at the beach to your friend, what features would you focus on?

One end of a rope is shaken to create a wave with a frequency of $0.20 \mathrm{~Hz}$. The rope is displaced with a maximum distance of $0.10 \mathrm{~m}$ away from its starting point. The speed of the wave through the rope is $10.0 \mathrm{~m} / \mathrm{s}$. A student graphs the wave. Complete the graphs by labeling the axes and write mathematical representations for the wave.

Two waves simultaneously present on a long string have a phase difference $\phi$ between them so that a standing wave formed from their combination is described by

$$y(x, t)=2 A \sin \left(k x+\frac{\phi}{2}\right) \cos \left(\omega t-\frac{\phi}{2}\right)$$

(a) Despite the presence of the phase angle $\phi$, is it still true that the nodes are one-half wavelength apart?

Two sources of harmonic waves on the $x$ axis have a phase difference that is proportional to time: $\delta_{\mathrm{s}}=\mathrm{Ct}$, where $C$ is a constant. The amplitude of the wave from each source at some point $P$ on the $x$ axis is $A_0$. ( $(a)$ Write the wave functions for each of the two waves at point $P$, assuming this point to be a distance $x_1$ from one source and $x_1+\Delta x$ from the other. (b) Find the resultant wave function and show that its amplitude is $2 A_0 \cos \left[\frac{1}{2}\left(\delta+\delta_0\right)\right]$, where $\delta$ is the phase difference at $P$ due to the path difference. (c) Using a spreadsheet program or graphing calculator, graph the intensity at point $P$ versus time for a zero path difference. (Let $I_0$ be the intensity due to each wave separately.) What is the time average of the intensity? $(d)$ Make the same graph for the intensity at a point for which the path difference is $\lambda / 2$.

Now try going the other way! Given the two graphs shown, determine the wave properties and write distance and time mathematical representations for the wave. What is the speed of this wave?

A coach shouts "Go!" from the finish line of a 100.0-m track to the runners. It takes $292 \mathrm{~ms}$ for the 171-Hz sound wave to reach them. Calculate the speed of sound in air and the wavelength of her voice.

You are using sonar over a spot in the ocean known to be $91.8 \mathrm{~m}$ deep. The system emits a $50.0-\mathrm{kHz}$ pulse into the water. It takes $120.0 \mathrm{~ms}$ for the sound to travel to the ocean floor and back to the sonar system. Determine the speed of sound in the ocean and the wavelength of the sonar pulse.

What causes a change in wave speed? What causes a change in frequency? Can you change the frequency of a wave to cause a change in the speed? How can you affect the wavelength of a wave?

The distance between the peaks of water waves seems to depend on how close they are to the shore, as shown in the image. Qualitatively describe the pattern that you observe.

Three electromagnetic waves travel through a certain point P along an x axis. They are polarized parallel to a y axis, with the following variations in their amplitudes. Find their resultant at P.

$$E_1 = (10.0 μV/m) sin[(2.0 × 10^14 rad/s)t]$$

$$E_2 = (5.00 μV/m) sin[(2.0 × 10^14 rad/s)t + 45.0°]$$

$$E_3 = (5.00 μV/m) sin[(2.0 × 10^14 rad/s)t - 45.0°]$$