Question

A 1.50-m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 48.0 m/s. What are the wavelength and frequency of (a) the fundamental tone; (b) the second overtone; (c) the fourth harmonic?

Answer 1

Answer: a) 16Hz, 3m b) 48Hz, 1mc) 80Hz, 0.6m

Explanation:

a) Fundamental frequency in string is represented as Fo = V/2L where;

Fo is the fundamental frequency

V is the speed of the transverse wave = 48m/s

L is the length of the wire. = 1.50m

Substituting this values in the formula given we have;

Fo = 48/2(1.5)

Fo = 48/3

Fo = 16Hz

The fundamental tone is therefore 16Hz

Using v =f¶

Where f is the frequency and ¶ is the wavelength, the wavelength of the fundamental note will be;

¶ = v/fo

¶ = 48/16 = 3m

b) Overtones or harmonics is the multiple integral of the fundamental frequency. The multiples are I'm arithmetical progression.

First overtone f1 = 2fo

Second overtone f2 = 3fo etc.

Since fo = 16Hz

Second overtone f2 = 3×16 = 48Hz

¶ = v/f2 = 48/48

¶ = 1m

c) Fourth harmonic or overtone will be f4 = 5fo

F4 = 5×16 = 80Hz

The fourth harmonic is therefore 80Hz

¶ = v/f4 = 48/80

¶ = 0.6m