Question

A periodic wave can be mathematically written as ???? = ???? sin(???????? − ????????) 1. What part of the wave does A describe? 2. If we take a snapshot of the wave, what is the distance between two crests (or troughs) called? How is this related to k? 3. The source that is generating the wave oscillates with a frequency of f . How is this related to ????? 4. Write down the relationship between wave speed, wave frequency and wavelength.

Answer 1

A periodic wave can be written as:

$y(x,t)=A sin (kx-\omega t)$

where

A is the amplitude, k is the wave number, x is the position, $\omega$ is the angular frequency and t is the time.

1. A represents the amplitude

Explanation: the term A inside the wave equation represents the amplitude, which is the maximum displacement of the wave (along the y-axis) with respect to tis equilibrium position. Basically, when the sine part of the wave is equal to 1, y=A, and the wave has the maximum displacement.

2. Wavelength

Explanation: the wavelength of a wave is defined as the distance between two successive crests (or between succesive throughs) of the wave.

The wavelength (indicated with $\lambda$) is related to the wave number, k, by the equation

$k=\frac{2\pi}{\lambda}$

3. $\omega = 2 \pi f$

The frequency at which the source of the wave is oscillating, f, is related to the angular frequency $\omega$ by the relationship

$\omega = 2 \pi f$

basically, f expresses the frequency in units of Hertz (1/s), while the angular frequency expresses the frequency in units of radians per second.

4. $v=f \lambda$

The relationship between wave speed, wave frequency and wavelength is

$v=f \lambda$

where

v is the wave speed

f is the frequency

$\lambda$ is the wavelength

We can observed that if the speed of the wave is constant, then the frequency and the wavelength are inversely proportional to each other.