The constituent waves of a stationary wave have amplitude frequency and velocity as 8 cm, 25 Hz and 150 cm `s^(-1)` respectively. What is the amplitude of the stationary wave at x=2cm

Explanation

To find the amplitude of the stationary wave at \( x = 2 \, \text{cm} \), we will follow these steps:

### Step 1: Identify Given Values
We have the following values:
- Amplitude \( a = 8 \, \text{cm} \)
- Frequency \( f = 25 \, \text{Hz} \)
- Velocity \( v = 150 \, \text{cm/s} \)

### Step 2: Calculate Angular Frequency \( \omega \)
The angular frequency \( \omega \) is calculated using the formula:
\[ \omega = 2\pi f \]
Substituting the value of \( f \):
\[ \omega = 2\pi \times 25 = 50\pi \, \text{rad/s} \]

### Step 3: Calculate Wave Number \( k \)
The wave number \( k \) is calculated using the formula:
\[ k = \frac{\omega}{v} \]
Substituting the values of \( \omega \) and \( v \):
\[ k = \frac{50\pi}{150} = \frac{\pi}{3} \, \text{rad/cm} \]

### Step 4: Write the Equation of the Stationary Wave
The equation of the stationary wave can be expressed as:
\[ y = 2a \sin(kx) \cos(\omega t) \]
Substituting \( a = 8 \, \text{cm} \):
\[ y = 2 \times 8 \sin\left(\frac{\pi}{3} x\right) \cos(50\pi t) \]
This simplifies to:
\[ y = 16 \sin\left(\frac{\pi}{3} x\right) \cos(50\pi t) \]

### Step 5: Substitute \( x = 2 \, \text{cm} \) into the Equation
Now, we will find the amplitude at \( x = 2 \, \text{cm} \):
\[ y = 16 \sin\left(\frac{\pi}{3} \times 2\right) \cos(50\pi t) \]
Calculating \( \frac{\pi}{3} \times 2 \):
\[ \frac{\pi}{3} \times 2 = \frac{2\pi}{3} \]
Thus, we have:
\[ y = 16 \sin\left(\frac{2\pi}{3}\right) \cos(50\pi t) \]

### Step 6: Calculate \( \sin\left(\frac{2\pi}{3}\right) \)
Using the known value:
\[ \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} \]
Substituting this value back into the equation:
\[ y = 16 \times \frac{\sqrt{3}}{2} \cos(50\pi t) \]
This simplifies to:
\[ y = 8\sqrt{3} \cos(50\pi t) \, \text{cm} \]

### Final Result
The amplitude of the stationary wave at \( x = 2 \, \text{cm} \) is:
\[ \boxed{8\sqrt{3} \, \text{cm}} \]