Question

Three resonant frequencies of a string are 90, 150 an 210 Hz. (a) Find the highest possible funclamera frequency of vibration of this string. (b) Which harmonics of the fundamental are the given frequencies ? (c) Which overtones are these frequencies ? (d) If the length of the string is 80 cm, what would be the speed of a transverse wave on this string?

Answer 1

The resonant frequencies of a string are

$f_1=90Hz$

$f_2=150Hz$

$f_3=210Hz$

(a) The highest possible fundamental frequency of the string is

f = 30 Hz

[because $f_1,f_2$ and $f_3$ are integral multiples of 30 Hz]

(b) The frequencies are

$f_1=3f$

$f_2=5f$

$f_3=7f$

So, $f_1,f_2$ and $f_3$ are $3^{rd}$ harmonic, $5^{th}$ harmonic and $7^{th}$ harmonic respectively.

(c) The frequency in the string are f, 2f, 3f, 4f, 5,......

So, 3f = 2nd overtone and 3rd harmonic

5f = 4th overtone and 5th harmonic

5th = 6th overtone and 7th harmonic

(d) Length of the string is,

L = 80 cm

$\Rightarrow f_1=\left(\frac{3}{2}\right)v$

(v = velocity of the wave)

$\Rightarrow 90=\{\frac{3}{(2\times 80)}\times K\}$

$\Rightarrow K=\frac{(90\times 2\times 80)}{3}$

$=30\times 2\times 80$

$4800cm/sec$

$48m/s$