Question

If ${\mathrm{\lambda }}_1,{\mathrm{\lambda }}_2,{\mathrm{\lambda }}_3$​ are the wavelengths of the waves giving resonance in the fundamental, first and second overtone modes respectively in a open organ pipe, then the ratio of the wavelengths ${\mathrm{\lambda }}_1:{\mathrm{\lambda }}_2:{\mathrm{\lambda }}_3$​ is?

Answer 1

Step 1. Given data

From the given, ${\mathrm{\lambda }}_1,{\mathrm{\lambda }}_2,{\mathrm{\lambda }}_3$ are wavelengths.

We have to find the ratio of the given wavelengths.

Step 2. Concept used

1. The distance between identical points on consecutive waves is known as wavelength.
2. The maximum displacement or distance moved by a point on a vibrating body or wave is measured from its equilibrium position.
3. Frequency is the number of occurrences of a repeating event per unit of time.

Step 3. Formula to be used

We know that frequency of overtone is,

$\mathrm{f}=\frac{\mathrm{n\lambda }}{2\mathrm{l}}$

Here, $\mathrm{f}$ is the frequency and $\mathrm{l}$ is the length.

Step 4. Find the wavelengths

The fundamental overtone mode is $\mathrm{n}=1$.

And the frequency is, ${\mathrm{f}}_0=\frac{\mathrm{\lambda }}{2\mathrm{l}}$

For first overtone mode is $\mathrm{n}=2$.

The frequency is, ${\mathrm{f}}_1=2\left(\frac{\mathrm{\lambda }}{2\mathrm{l}}\right)$

For second overtone mode is $\mathrm{n}=3$,

And its frequency is, ${\mathrm{f}}_2=3\left(\frac{\mathrm{\lambda }}{2\mathrm{l}}\right)$

So,

${\mathrm{f}}_0:{\mathrm{f}}_1:{\mathrm{f}}_2=1:2:3$

Step 5. Find the ratio of wavelengths

We know that,

$\mathrm{f}=\frac{\mathrm{v}}{\mathrm{\lambda }}$

Here, $\mathrm{v}$ is the velocity.

So,

$\frac{\mathrm{v}}{{\mathrm{\lambda }}_1}:\frac{\mathrm{v}}{{\mathrm{\lambda }}_2}:\frac{\mathrm{v}}{{\mathrm{\lambda }}_3}=1:2:3$

Here, $\mathrm{v}$ is constant.

Therefore,

$\frac{1}{{\mathrm{\lambda }}_1}:\frac{1}{{\mathrm{\lambda }}_2}:\frac{1}{{\mathrm{\lambda }}_3}=1:2:3$

${\mathrm{\lambda }}_1:{\mathrm{\lambda }}_2:{\mathrm{\lambda }}_3=1:\frac{1}{2}:\frac{1}{3}$

Hence, the ratio of the wavelengths ${\mathrm{\lambda }}_1:{\mathrm{\lambda }}_2:{\mathrm{\lambda }}_3$ are $1:\frac{1}{2}:\frac{1}{3}$.