## 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

- 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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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(18*x* – 5.2*t*).

Practice 14.2.6 |
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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.

**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?