A string 2.0 m long and fixed at its ends is driven by a 240 Hz vibrator. The string vibrates in its third harmonic mode. The speed of the wave and its fundamental frequency is:

Concept:

Wave speed:

The speed of the wave is given by,

Wave speed = frequency × wavelength.

 v = fλ

Wavelength (λ) is the distance between two corresponding points on adjacent waves. Wave frequency (f) is the number of waves that pass a fixed point in a given amount of time.

Fundamental Frequency:

The lowest frequency of any vibrating object is called the fundamental frequency.

i.e. the lowest frequency which is produced by the oscillation i.e., the fundamental frequency is given as,

\({f_1} = \frac{f}{n}\)

Calculation:

Given,

Frequency, f = 240 Hz

Length of the string, l = 2 m

The string vibrates in third harmonic mode i.e., n = 3

Standing waves of many different wavelengths can be produced on a string with two fixed ends, as long as an integral number of half wavelengths fits into the length of the string.

For a standing wave on a string of length L with two fixed ends, its wavelength will be

\(\lambda = \frac{{2l}}{n}\)

\(\lambda = 2 \times \frac{2}{3} = \frac{4}{3}\)

\(\lambda = \frac{4}{3}m\)

The speed of the wave is given by,

Velocity, \(v = f\lambda = 240 \times \frac{4}{3} = 320\) m/sec

By fundamental frequency formula,

\({f_1} = \frac{f}{n}\)

\({f_1} = \frac{{240}}{3} = 80\;{\rm{Hz}}\)