16.2 Mathematics of Waves

The wave function above is derived using a sine function. Can a cosine function be used instead?

The wave equation $\frac{\partial^{2} y \left(x , t\right)}{\partial x^{2}} = \frac{1}{v^{2}} \frac{\partial^{2} y \left(x , t\right)}{\partial t^{2}}$ works for any wave of the form $y \left(x , t\right) = f \left(x +- v t\right) .$ In the previous section, we stated that a cosine function could also be used to model a simple harmonic mechanical wave. Check if the wave

is a solution to the wave equation.

If you shake the end of a stretched spring up and down with a frequency f, you can produce a sinusoidal, transverse wave propagating down the spring. Does the wave number depend on the frequency you are shaking the spring?

In this section, we have considered waves that move at a constant wave speed. Does the medium accelerate?

A transverse wave on a string is modeled with the wave function $y \left(x , t\right) = \left(0.20 \text{cm}\right) \text{sin} \left(2.00 \text{m}^{−1} x - 3.00 \text{s}^{−1} t + \frac{\pi}{16}\right) .$ What is the height of the string with respect to the equilibrium position at a position $x = 4.00 \text{m}$ and a time $t = 10.00 \text{s} ?$

A wave is modeled with the function $y \left(x , t\right) = \left(0.25 \text{m}\right) \text{cos} \left(0.30 \text{m}^{−1} x - 0.90 \text{s}^{−1} t + \frac{\pi}{3}\right) .$ Find the (a) amplitude, (b) wave number, (c) angular frequency, (d) wave speed, (e) initial phase shift, (f) wavelength, and (g) period of the wave.

A wave is modeled by the wave function $y \left(x , t\right) = \left(0.30 \text{m}\right) \text{sin} \left[\frac{2 \pi}{4.50 \text{m}} \left(x - 18.00 \frac{\text{m}}{\text{s}} t\right)\right] .$ What are the amplitude, wavelength, wave speed, period, and frequency of the wave?

A swimmer in the ocean observes one day that the ocean surface waves are periodic and resemble a sine wave. The swimmer estimates that the vertical distance between the crest and the trough of each wave is approximately 0.45 m, and the distance between each crest is approximately 1.8 m. The swimmer counts that 12 waves pass every two minutes. Determine the simple harmonic wave function that would…

Consider two waves defined by the wave functions $y_{1} \left(x , t\right) = 0.50 \text{m} \text{sin} \left(\frac{2 \pi}{3.00 \text{m}} x + \frac{2 \pi}{4.00 \text{s}} t\right)$ and $y_{2} \left(x , t\right) = 0.50 \text{m} \text{sin} \left(\frac{2 \pi}{6.00 \text{m}} x - \frac{2 \pi}{4.00 \text{s}} t\right) .$ What are the similarities and differences between the two waves?

The speed of a transverse wave on a string is 300.00 m/s, its wavelength is 0.50 m, and the amplitude is 20.00 cm. How much time is required for a particle on the string to move through a distance of 5.00 km?