PHYS 2211 Module 14.2

Mathematics of Waves

Recommended Reading

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

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

Practice!

Practice 14.2.1
A sinusoidal, transverse wave that propagates in the positive x-direction can be described with the wave function , where k is the wave number, x is the position of a point on the wave, is the angular frequency, and t is time. What is another way to express the wave number, k?
(a)
(b)
(c)
(d)

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.

Practice 14.2.2
A sinusoidal, transverse wave that propagates in the positive x-direction can be described with the wave function , where k is the wave number, x is the position of a point on the wave, is the angular frequency, and t is time. What is another way to express the angular frequency, ?
(a)
(b)
(c)
(d)

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.

Practice 14.2.3
A wave traveling in a string can be described by the function where x is measured in meters, y is measured in centimeters, and t is measured in seconds. What is the speed of this wave?
(a) 4 m/s
(b) 8 m/s
(c) 0.5 m/s
(d) 2 m/s
(e) 3 m/s

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.

Pause & Predict 14.2.1
Which function correctly represents the wave in the string?
(a) y(x,t) = 0.091 sin(1.9x – 26t)
(b) y(x,t) = 0.091 sin(12x – 26t)
(c) y(x,t) = 0.091 sin(1.9x + 4.7t)
(d) y(x,t) = 0.091 sin(12x – 4.7t)
(e) y(x,t) = 0.091 sin(12x + 26t)

Pause & Predict 14.2.2
What is the speed of the wave?
(a) 13 m/s
(b) 5.2 m/s
(c) 2.3 m/s
(d) 32 m/s
(e) 0.18 m/s

Practice!

Practice 14.2.4
A wave traveling in a string has a wavelength of 35 cm, an amplitude of 8.4 cm, and a period of 1.2 s. What is the speed of this wave?
(a) 0.42 m/s
(b) 3.4 m/s
(c) 1.8 m/s
(d) 0.046 m/s
(e) 0.29 m/s

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.

Practice 14.2.5
A wave traveling in a string in the positive x direction has a wavelength of 35 cm, an amplitude of 8.4 cm, and a period of 1.2 s. What is the wave equation that correctly describes this wave?
(a) y(x,t) = 0.35 sin(18x – 5.2t)
(b) y(x,t) = 0.35 sin(0.35x – 1.2t)
(c) y(x,t) = 0.35 sin(2.9x – 0.83t)
(d) y(x,t) = 0.35 sin(18x + 5.2t)
(e) y(x,t) = 0.35 sin(0.35x + 1.2t)

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).

Practice 14.2.6
A transverse, sinusoidal wave travels in a string and can be described by the function: . What is the speed of this wave?
(a) 9.2 m/s
(b) 4.3 m/s
(c) 0.23 m/s
(d) 0.18 m/s

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.