A sound source of frequency 500 Hz is producing longitudinal waves in a spring.The distance between two consecutive rarefractions is 24 cm.If the amplitude of vibration of a particle of the spring is 3.0 cm and the wave is travelling in the negative x-direction, then write the equation for the wave.Assume that the source is at x=0 and at this point the displacement is zero at the time t=0.

Answer

To find the equation of the wave produced by a sound source of frequency 500 Hz, we can follow these steps:

### Step 1: Identify the Given Information
- Frequency (ν) = 500 Hz
- Distance between two consecutive rarefactions (λ) = 24 cm
- Amplitude (A) = 3.0 cm
- Wave direction = negative x-direction
- Displacement at x=0 and t=0 = 0

### Step 2: Calculate the Wavelength (λ)
The distance between two consecutive rarefactions is equal to the wavelength (λ) of the wave. Therefore, we have:
- λ = 24 cm = 0.24 m (converting to meters for standard SI units)

### Step 3: Calculate the Period (T)
The period (T) of the wave can be calculated using the frequency (ν):
\[ T = \frac{1}{ν} = \frac{1}{500 \text{ Hz}} = 0.002 \text{ s} \]

### Step 4: Write the General Equation of the Wave
The general equation for a wave traveling in the negative x-direction is given by:
\[ y(x, t) = A \cos\left(2\pi \left(\frac{t}{T} - \frac{x}{λ}\right)\right) \]

### Step 5: Substitute the Known Values into the Equation
Substituting the values of amplitude (A), period (T), and wavelength (λ) into the equation:
\[ y(x, t) = 3 \cos\left(2\pi \left(\frac{t}{0.002} - \frac{x}{0.24}\right)\right) \]

### Step 6: Simplify the Equation
Now, simplifying the argument of the cosine function:
\[ y(x, t) = 3 \cos\left(2\pi \cdot 500 t - \frac{2\pi}{0.24} x\right) \]

### Step 7: Final Equation
The final equation of the wave is:
\[ y(x, t) = 3 \cos\left(1000\pi t - \frac{25\pi}{3} x\right) \]

### Summary
The equation of the wave produced by the sound source is:
\[ y(x, t) = 3 \cos\left(1000\pi t - \frac{25\pi}{3} x\right) \]

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