Two plane harmonic sound waves are expressed by the equations
`y_(1)(x,t)=Acos((pi)/(2)x-100pit)`
`y_(2)(x,t)=Acos((46pi)/(100)x-92pit)`
All quantities taken in MKS.
Q. What is the speed of the sound?

Options

A. 200 m/s
B. 180 m/s
C. 192 m/s
D. 96 m/s

Answer

To find the speed of sound from the given equations of two plane harmonic sound waves, we can follow these steps:

### Step 1: Identify the General Form of the Wave Equation
The general equation for a harmonic sound wave can be expressed as:
\[ y(x, t) = A \cos(kx - \omega t) \]
where:
- \( A \) is the amplitude,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency.

### Step 2: Compare Given Equations with the General Form
The given equations are:
1. \( y_1(x, t) = A \cos\left(\frac{\pi}{2} x - 100 \pi t\right) \)
2. \( y_2(x, t) = A \cos\left(\frac{46\pi}{100} x - 92 \pi t\right) \)

From these, we can identify:
- For \( y_1 \):
- \( \omega_1 = 100 \pi \)
- \( k_1 = \frac{\pi}{2} \)

- For \( y_2 \):
- \( \omega_2 = 92 \pi \)
- \( k_2 = \frac{46\pi}{100} = \frac{23\pi}{50} \)

### Step 3: Calculate the Speed of Sound for Each Wave
The speed of sound \( v \) can be calculated using the formula:
\[ v = \frac{\omega}{k} \]

#### For Wave 1:
Using \( \omega_1 \) and \( k_1 \):
\[ v_1 = \frac{\omega_1}{k_1} = \frac{100 \pi}{\frac{\pi}{2}} = \frac{100 \pi \cdot 2}{\pi} = 200 \, \text{m/s} \]

#### For Wave 2:
Using \( \omega_2 \) and \( k_2 \):
\[ v_2 = \frac{\omega_2}{k_2} = \frac{92 \pi}{\frac{23\pi}{50}} = \frac{92 \pi \cdot 50}{23 \pi} = \frac{4600}{23} \approx 200 \, \text{m/s} \]

### Step 4: Conclusion
Both calculations yield the same speed of sound:
\[ v = 200 \, \text{m/s} \]

Thus, the speed of sound is:
\[ \boxed{200 \, \text{m/s}} \]

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