# Beats in Sound Waves

This topic is part of the HSC Physics course under the section Sound Waves.

### HSC Physics Syllabus

• analyse qualitatively and quantitatively the relationships of the wave nature of sound to explain:
– beats f_{\text{beat}}=|f_2−f_1|

### What are Beats?

Beats are the periodic and rhythmic fluctuations in the loudness or intensity of a sound. They occur when two sound waves of slightly different frequencies interfere with each other. This interference results in alternating constructive and destructive interference, leading to the characteristic "waxing and waning" or "throbbing" effect of beats.

### How Do Beats Form?

Imagine playing two tuning forks with frequencies that are close but not exactly the same. The sound waves produced by these tuning forks will sometimes align (constructive interference) leading to a louder sound, and at other times, they will be out of phase (destructive interference) leading to a softer sound or silence. When in phase, the compressions (regions of high pressure) of one wave align with  the compressions of the other, and the rarefactions (regions of low pressure) also align, resulting in an amplified sound.

When out of phase, the compression of one wave aligns with the rarefaction of the other, essentially cancelling each other out and producing a quiet or silent moment.

This alternating loud-soft-loud pattern is perceived as beats.

### Calculating Beat Frequency

The frequency of the beats is equal to the absolute difference between the frequencies of the two interfering sound waves. Mathematically, this can be expressed as:

$$f_{\text{beats}} = |f_1 - f_2|$$

Where:

• is the beat frequency.
• and f_2 are the frequencies of the two sound waves.

The frequency of the sound wave from interference is also the average of the waves' frequencies.

$$f_{\text{resultant}} = \frac{f_1 + f_2}{2}$$

### Example

A sound wave at a frequency of 360 Hz produces 32 beats in 4 seconds when interacting with a vibrating tuning fork.

(a) What is the beat frequency?

(b) What are the two possible frequencies of the vibrating tuning fork?

(c) What is the frequency of the resultant wave formed from interference?

Solution for part (a):

32 beats in 4 seconds is equivalent to 8 beats in 1 second (beat frequency).

Solution for part (c):

The frequency of one of the sound waves is 360 Hz which means the vibrating tuning fork must have a frequency that is either 8 Hz higher or lower than 360 Hz.

Therefore:

$$f_{\text{beats}} = |f_1 - f_2|$$

$$8 = |360 - f_2|$$

$$f_2 = 352 \; \text{or} \, 368 \, \text{Hz}$$

Solution for part (b):

The frequency of the resultant wave depends on the frequency of the tuning fork.

If the tuning fork's frequency is 352 Hz, then the resultant wave's frequency is

$$f = \frac{352 + 360}{2} = 356 \, \text{Hz}$$

If the tuning fork's frequency is 368 Hz, then the resultant wave's frequency is

$$f = \frac{368 + 360}{2} = 364 \, \text{Hz}$$

### Applications and Relevance

Beats have practical applications in various fields:

1. Musical Instruments: Musicians often use beats to tune their instruments. By comparing a note of known frequency with the note produced by the instrument, they can adjust the instrument until the beat frequency is zero, indicating matching frequencies.
2. Acoustics and Audio Engineering: Understanding beats is essential for sound engineers and producers to ensure the clarity of audio recordings and avoid unwanted interference.