Question

Three resonant frequencies of the string are $90,150and210Hz$.

1. Find the highest possible fundamental frequency of vibration of this string.
2. Which harmonics of the fundamental are the given frequencies?
3. Which overtones are these frequencies?
4. If The length of the string is $80cm$, what would be the speed of a transverse wave on this string?

Answer 1

Step 1: Calculate for (a),

The greatest common number of $90,150and210is30$

So, $f=30Hz$

Step 2: Calculate for or (b),

Let$f_1,f_{2}and{f}_3$ are Three resonant frequencies of the string.

$f_1=3f,f_2=5fand{f}_3=7f$

Step 3: Calculate for (c),

$n^{th}$ overtone is ${(n+1)}^{th}$ frequency.

So,$3fis{2}^{nd}$ overtone and $3^{rd}$ harmonic.

$5fis{4}^{th}$ overtone and $6^{th}$ harmonic.

and$7fis{6}^{th}$ overtone and $7^{th}$harmonic.

Step 4: Calculate for (d),

Let, $f_1=\left(\frac{3}{2}\right)LKwhereLislength=80cmgiven,Kisvelocity$

so, $90=(\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\times 80)k$

$\Rightarrow k=48\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$.