Waves Physics Example 4

Follow the full solution, then compare it with the other examples linked below.

hard

A wave on a string has the equation

y(x,t) = 0.05 \sin(4\pi x - 8\pi t)

(SI units). What are the amplitude, wavelength, frequency, and wave speed?

1
Standard form: $y = A\sin(kx - \omega t)$ . So $A = 0.05 \text{ m}$ , $k = 4\pi \text{ rad/m}$ , $\omega = 8\pi \text{ rad/s}$ .
2
Wavelength: $\lambda = \frac{2\pi}{k} = \frac{2\pi}{4\pi} = 0.5 \text{ m}$ . Frequency: $f = \frac{\omega}{2\pi} = \frac{8\pi}{2\pi} = 4 \text{ Hz}$ .
3
Wave speed: $v = f\lambda = 4 \times 0.5 = 2 \text{ m/s}$ (or $v = \frac{\omega}{k} = \frac{8\pi}{4\pi} = 2 \text{ m/s}$ ).

Answer

A = 0.05 \text{ m}, \; \lambda = 0.5 \text{ m}, \; f = 4 \text{ Hz}, \; v = 2 \text{ m/s}

A sinusoidal wave equation encodes all wave properties. The wave number

k

and angular frequency

\omega

relate to wavelength and frequency through factors of

2\pi

About Waves

A disturbance that transfers energy and information through space or a medium without permanently displacing the matter it travels through.

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