# Two ultrasonic sound waves combine and form a beat frequency that is in the range of human...

## Question:

Two ultrasonic sound waves combine and form a beat frequency that is in the range of human hearing for a healthy young person. The frequency of one of the ultrasonic waves is {eq}84 \ kHz {/eq}. (The range of hearing for a healthy young person is {eq}20 \ Hz {/eq} to {eq}20 \ kHz {/eq}.)

(a) What is the smallest possible value for the frequency of the other ultrasonic wave?

(b) What is the largest possible value for the frequency of the other ultrasonic wave?

## Interference in Sound

Two coherent light beams interfere to constructive and destructive interference patterns or alternate dark and bright bands. In the similar manner sound waves also interfere each other. The interfered sound produces alternate maxima and minima. Normally the listener hears only the maxima. A pair of maxima and minima together in interference of sound is termed as one beat. Number of beats heard per second will be equal to the difference in frequency between the two interfering waves. If {eq}F_1, \ \ F_2 {/eq} are the interfering sound frequencies the beat frequency {eq}F_b = | F_1 - F_2| {/eq}

Given data

• One frequency of the Ultrasonic sound {eq}F_1 = 84 \times 10^3 \ Hz {/eq}
• Range of frequency of hearing for a healthy person {eq}20 \ Hz \ \ to \ \ 20 \times 10^3 \ Hz {/eq}

Part a )

The beat frequency produced must be with in the range of hearing. The smallest hearing frequency is 20 kHz

So when the smallest frequency ultra sound must produce a frequency greater than or equal to 20k Hz

Let {eq}F_2 {/eq} be the smallest ultra sound that can produce a hear able beat

Then we have the equation for beat {eq}F_b = F_1 - F_2 {/eq}

Then the smallest ultra sound frequency {eq}F_2 = F_1 - F_b \\ F_2 = 84 \times 10^3 - 20 \times 10^3 \\ F_2 = 64 \times 10^3 \ \ Hz {/eq}

Part b)

Let {eq}F_3 {/eq} be the highest possible frequency.

Then the highest frequency {eq}F_3 = F_1+ F_b \\ F_3 = 84 \times 10^3 + 20 \times 10^3 \\ F_3 = 104 \times 10^3 \ Hz {/eq}