Beats: Videos & Practice Problems

Beats occur when two sound waves with similar frequencies interfere, creating an oscillation in amplitude. For example, if one wave has a frequency of 8 Hz and another 10 Hz, the beat frequency is calculated as the absolute value of their difference, resulting in 2 Hz. The resulting sound has a pitch equal to the average frequency of the two waves, while the loudness varies at the beat frequency. Understanding this phenomenon involves concepts like constructive and destructive interference, amplitude, and frequency, which are essential in acoustics.

Beats

Waves can interfere with each other, creating unique patterns, and one fascinating phenomenon is known as beats, which occurs with sound waves. To understand beats, consider two sound waves, referred to as wave A and wave B, with frequencies of 8 Hertz and 10 Hertz, respectively. When these waves are played simultaneously, they interfere with each other, leading to alternating patterns of constructive and destructive interference. Constructive interference occurs when the waves align perfectly, resulting in increased amplitude, while destructive interference happens when the waves cancel each other out, leading to zero displacement.

The resulting wave from this interference is characterized by a varying amplitude, which oscillates between a maximum value and zero. This oscillation in amplitude is what defines a beat. The amplitude of the combined wave is effectively the sum of the amplitudes of the individual waves. For instance, if both waves have an amplitude of 1, the resulting wave will have an amplitude of 2.

To quantify the beat frequency, which is the number of beats heard per second, one can use the formula:

f_{beats} = |f_A - f_B|

In this case, with frequencies of 8 Hz and 10 Hz, the beat frequency is calculated as:

f_{beats} = |8 - 10| = 2 \text{ Hz}

This indicates that the listener would perceive two complete oscillations of amplitude in one second.

Additionally, the pitch of the sound produced by the combined waves is determined by the average of the two frequencies:

f_{sound} = \frac{f_A + f_B}{2} = \frac{8 + 10}{2} = 9 \text{ Hz}

While the loudness varies according to the beat frequency, the pitch remains constant at this average frequency.

To illustrate this concept further, consider two musicians playing notes with wavelengths of 0.65 meters and 0.654 meters, respectively. Given the speed of sound at 343 m/s, the frequencies can be calculated using the relationship:

v = \lambda f

For wave A:

f_A = \frac{v}{\lambda_A} = \frac{343}{0.65} \approx 527.7 \text{ Hz}

For wave B:

f_B = \frac{v}{\lambda_B} = \frac{343}{0.654} \approx 524.5 \text{ Hz}

The beat frequency is then determined by:

f_{beats} = |f_A - f_B| = |527.7 - 524.5| \approx 3.2 \text{ Hz}

This means the musicians would hear a beat oscillation approximately 3.2 times per second, creating a distinct auditory experience.

Here’s what students ask on this topic:

What are beats in sound waves?

Beats in sound waves occur when two sound waves with slightly different frequencies interfere with each other. This interference results in an oscillation in the amplitude of the resultant wave. The beat frequency, which is the rate at which the amplitude oscillates, is equal to the absolute difference between the two frequencies. For example, if one wave has a frequency of 8 Hz and another has 10 Hz, the beat frequency is |10 - 8| = 2 Hz. The pitch of the resulting sound is the average of the two frequencies, while the loudness varies at the beat frequency.

How do you calculate the beat frequency?

The beat frequency is calculated using the absolute value of the difference between the frequencies of the two interfering sound waves. The formula is:

$f{\mathrm{beat}}_{}=|f{1}_{}-f{2}_{}|$

For example, if the frequencies of the two waves are 8 Hz and 10 Hz, the beat frequency is |10 - 8| = 2 Hz. This means you will hear 2 beats per second.

What is the difference between beat frequency and the resulting sound frequency?

The beat frequency is the rate at which the amplitude of the sound oscillates due to interference, calculated as the absolute difference between the two frequencies. The resulting sound frequency, or pitch, is the average of the two frequencies. For example, if you have two waves with frequencies 8 Hz and 10 Hz, the beat frequency is |10 - 8| = 2 Hz, and the resulting sound frequency is (8 + 10) / 2 = 9 Hz. The beat frequency affects the loudness variation, while the resulting sound frequency determines the pitch.

How do constructive and destructive interference relate to beats?

Constructive and destructive interference are key to understanding beats. Constructive interference occurs when the crests and troughs of two waves align, resulting in a higher amplitude. Destructive interference happens when the crest of one wave aligns with the trough of another, canceling each other out and resulting in a lower amplitude. In beats, these interferences alternate, causing the amplitude of the resultant wave to oscillate. This oscillation in amplitude is what we perceive as beats, with the beat frequency being the rate of these oscillations.

What would the beat frequency be if two sound waves have frequencies of 440 Hz and 442 Hz?

To find the beat frequency, you use the formula:

$f{\mathrm{beat}}_{}=|f{1}_{}-f{2}_{}|$

For frequencies of 440 Hz and 442 Hz, the beat frequency is:

$|442-440|=2 \mathrm{Hz}$

This means you will hear 2 beats per second.