MP Board · Class 9 · Science · Sound(a) Define the following terms related to a sound wave: Amplitude Wavelength Frequency Time Period (b) Derive the mathematical relation between wave speed ($v$), frequency ($\nu$), and wavelength ($\lambda$). (c) A sound wave has a frequency of $2\text{ kHz}$ and a wavelength of $35\text{ cm}$. Calculate the time it will take to travel a distance of $1.4\text{ km}$.

(a) Definitions of Sound Wave Characteristics

1. Amplitude ($A$): The maximum displacement of the particles of the medium from their original mean position on either side as the sound wave passes. It determines the loudness or volume of the sound.
2. Wavelength ($\lambda$): The distance between two consecutive compressions or two consecutive rarefactions in a sound wave. Its SI unit is the meter ($\text{m}$).
3. Frequency ($\nu$): The number of complete wave cycles or oscillations produced per unit time (per second). Its SI unit is Hertz ($\text{Hz}$).
4. Time Period ($T$): The time required for a wave to complete one full oscillation or cycle. Its SI unit is the second ($\text{s}$).

(b) Derivation of Relationship between Speed, Frequency, and Wavelength\nBy definition, wave speed is the distance travelled by the wave per unit time:

$$\text{Wave Speed } (v) = \frac{\text{Distance travelled}}{\text{Time taken}}$$ \nFor one complete wave cycle:

• The distance travelled is equal to the wavelength ($\lambda$).
• The time taken is equal to the time period ($T$). \nSubstituting these into the speed formula: $$v = \frac{\lambda}{T}$$ \nSince frequency ($\nu$) is the reciprocal of the time period ($T$), i.e., $\nu = \frac{1}{T}$: $$v = \lambda \times \left(\frac{1}{T}\right) = \lambda \times \nu$$

$$\mathbf{v = \nu \times \lambda}$$ $$\text{Wave Speed} = \text{Frequency} \times \text{Wavelength}$$

(c) Step-by-Step Numerical Solution

Given:

• Frequency, $\nu = 2\text{ kHz} = 2 \times 1000\text{ Hz} = 2000\text{ Hz}$
• Wavelength, $\lambda = 35\text{ cm} = \frac{35}{100}\text{ m} = 0.35\text{ m}$
• Distance, $s = 1.4\text{ km} = 1.4 \times 1000\text{ m} = 1400\text{ m}$

Step 1: Calculate the speed of the sound wave ($v$) $$v = \nu \times \lambda$$ $$v = 2000\text{ Hz} \times 0.35\text{ m} = 700\text{ m/s}$$

Step 2: Calculate the time taken ($t$) to travel $1400\text{ m}$ $$\text{Time } (t) = \frac{\text{Distance } (s)}{\text{Speed } (v)}$$ $$t = \frac{1400\text{ m}}{700\text{ m/s}} = 2\text{ s}$$

Answer: The sound wave will take $2\text{ seconds}$ to travel a distance of $1.4\text{ km}$.