# PHYS 2211 Module 14.2

## Mathematics of Waves 14.2 Mathematics of Waves

### Learning Objectives

By the end of this section, you will be able to:

• Model a wave, moving with a constant wave velocity, with a mathematical expression
• Calculate the velocity and acceleration of the medium
• Show how the velocity of the medium differs from the wave velocity (propagation velocity)

### Modeling a One-Dimensional Sinusoidal Wave using a Wave Function

The wave function modeling a sinusoidal wave, allowing for an initial phase shift 𝜙, is:

The value is known as the phase of the wave, where 𝜙 is the initial phase of the wave function. Whether the temporal term 𝜔𝑡 is negative or positive depends on the direction of the wave.

### Finding the Characteristics of a Sinusoidal Wave

1. To find the amplitude, wavelength, period, and frequency of a sinusoidal wave, write down the wave function in the form .
2. The amplitude can be read straight from the equation and is equal to A.
3. The period of the wave can be derived from the angular frequency: .
4. The frequency can be found using 5. The wavelength can be found using the wave number:  Practice!

Another way to express the wave number is where is the wavelength. The wave number is just a convenient quantity to use in a wave function. It has units of rad/m.

Another way to express the angular frequency is where f is the frequency. The angular frequency, , is measured in units of rad/s and the frequency, f, is measured in units of Hz, or cycles per second. Since there are radians per cycle, this is a way to convert between angular frequency and frequency.

The speed of this wave is 8 m/s. This wave function describes a wave with wave number k = 0.5 rad/m and angular frequency = 4 rad/s. The speed of the wave can be calculated by taking the angular frequency divided by the wave number: , which gives = 8 m/s.

Alternatively, the wavelength can be calculated from the wave number, where . From this wave function the wavelength is . The frequency can be calculated from the angular frequency, where . From this wave function, the frequency is . The speed of the wave can be calculated with the product of the wavelength and the frequency, where = 8 m/s. Practice!

The speed of any wave is where is the wavelength and T is the period. The wave speed for this wave is = 0.29 m/s.

The general form of a wave equation is , where A is the amplitude, the wave number is = 18 m-1, and the angular frequency is = 5.2 rad/s. The argument of the sine function is for a wave traveling in the positive x direction. So, the wave equation is y(x,t) = 0.35 sin(18x – 5.2t).

The speed of a wave is where is the wavelength, f is the frequency, is the angular frequency, and k is the wave number. For this wave, k is 21 m-1 and is 4.9 rad/s, so the wave speed is = 0.23 m/s. Discuss!

Two traveling waves 1 and 2 are described by the equations:

(a) Which wave has the higher speed?

(b) What is the wavelength of wave 1?

(c) What is the period of wave 2?