8. Deriving relationship between speed, frequency and wavelength

Question:

5 m.

Answer the following in brief:

(a) Derive the relation between speed, wavelength and frequency of a sound wave.

(b) A sound wave travels a distance of $340\ m$ in $1\ s$ in air. If its frequency is $343\ Hz$, calculate its wavelength.

(a) Step 1: Distance travelled in one complete vibration

1. What distance does a sound wave travel during one complete vibration?

$T$
$\frac{1}{\lambda}$
$\nu$
$lambda$

Step 2: Time taken for one complete vibration

The time taken to complete one vibration is called the time period $T$.

2. Which expression correctly represents the speed of the sound wave using wavelength and time period?

$v=\frac{T}{\nu}$
$v=\lambda T$
$v=\frac{T}{\lambda}$
$v=\frac{\lambda}{T}$

Step 3: Relation between frequency and time period

Frequency $\nu$ is the number of complete vibrations made in one second. Therefore, frequency and time period are related by:

3. If the frequency of a sound wave is $\nu$, which expression gives its time period $T$?

$T=v \nu$
$T=\lambda \nu$
$T=\frac{1}{\nu}$
$T= \nu$

Step 4: Substitute the value of $T$

$4.$ After substituting $T=\frac{1}{\nu}$ in $v=\frac{\lambda}{T}$, which expression is obtained?

$v=\frac{\lambda}{\nu} $
$v=\lambda +\nu$
$v=\frac{\lambda}{1/\nu}$
$v=\frac{\nu}{\lambda}$

Step 5: Final relation

5. Which is the final relation between speed, wavelength and frequency of a sound wave?

$v=\frac{\nu}{\lambda}$
$v=\lambda +\nu$
$v=\frac{\lambda}{\nu}$
$v=\lambda \nu$

(b) The wavelength of the sound wave $=$ $m$

[Note: Enter the answer upto two decimal values]

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