Two sinusoidal waves travelling in opposite directions interfere to produce a standing wave described by the equation
`y=(1.5m)sin (0.400x) cos(200t)`
where, x is in meters and t is in seconds. Determine the wavelength, frequency and speed of the interfering waves.

Explanation

To solve the problem, we need to extract the wavelength, frequency, and speed of the interfering waves from the given standing wave equation:

**Given Equation:**
\[ y = (1.5 \, \text{m}) \sin(0.400x) \cos(200t) \]

### Step 1: Identify the wave number (k)
The given standing wave equation can be compared with the standard form:
\[ y = A \sin(kx) \cos(\omega t) \]

From the equation, we can identify:
- \( k = 0.400 \, \text{m}^{-1} \)

### Step 2: Calculate the wavelength (λ)
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]

Rearranging this gives:
\[ \lambda = \frac{2\pi}{k} \]

Substituting the value of \( k \):
\[ \lambda = \frac{2\pi}{0.400} \]

Calculating this:
\[ \lambda = \frac{2\pi}{0.400} = 5\pi \, \text{m} \]

### Step 3: Identify the angular frequency (ω)
From the equation, we can also identify:
- \( \omega = 200 \, \text{s}^{-1} \)

### Step 4: Calculate the frequency (f)
The angular frequency \( \omega \) is related to the frequency \( f \) by the formula:
\[ \omega = 2\pi f \]

Rearranging gives:
\[ f = \frac{\omega}{2\pi} \]

Substituting the value of \( \omega \):
\[ f = \frac{200}{2\pi} \]

Calculating this:
\[ f = \frac{200}{2\pi} = \frac{100}{\pi} \, \text{Hz} \]

### Step 5: Calculate the speed (v)
The speed \( v \) of the wave is given by the product of frequency and wavelength:
\[ v = f \cdot \lambda \]

Substituting the values we found:
\[ v = \left(\frac{100}{\pi}\right) \cdot (5\pi) \]

Calculating this:
\[ v = 100 \cdot 5 = 500 \, \text{m/s} \]

### Final Results
- Wavelength \( \lambda = 5\pi \, \text{m} \)
- Frequency \( f = \frac{100}{\pi} \, \text{Hz} \)
- Speed \( v = 500 \, \text{m/s} \)

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