A wave is represented by the equation :` y = A sin(10 pi x + 15 pi t + pi//3) ` where, x is in metre and t is in second. The expression represents.

Options

A. a wave travelling in the positive x-direction with a velocity 1.5m/s
B. a wave travelling in the negative x-direction with a velocity 1.5m/s
C. a wave travelling in the negative x-direction with a wavelength 0.2m
D. a wave travelling in the positive x-direction with a wavelength 0.2m

Answer

To analyze the wave represented by the equation \( y = A \sin(10\pi x + 15\pi t + \frac{\pi}{3}) \), we can break down the components of the equation step by step.

### Step 1: Identify the General Form of the Wave Equation
The general form of a wave equation can be expressed as:
\[ y = A \sin(kx + \omega t + \phi) \]
where:
- \( A \) is the amplitude of the wave,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase constant.

### Step 2: Compare with the Given Equation
From the given equation:
\[ y = A \sin(10\pi x + 15\pi t + \frac{\pi}{3}) \]
we can identify:
- \( k = 10\pi \)
- \( \omega = 15\pi \)
- \( \phi = \frac{\pi}{3} \)

### Step 3: Calculate the Wave Number \( k \)
The wave number \( k \) is given by:
\[ k = \frac{2\pi}{\lambda} \]
where \( \lambda \) is the wavelength. We can rearrange this to find the wavelength:
\[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{10\pi} = \frac{1}{5} \text{ m} \]

### Step 4: Calculate the Angular Frequency \( \omega \)
The angular frequency \( \omega \) is related to the frequency \( f \) by:
\[ \omega = 2\pi f \]
We can rearrange this to find the frequency:
\[ f = \frac{\omega}{2\pi} = \frac{15\pi}{2\pi} = \frac{15}{2} = 7.5 \text{ Hz} \]

### Step 5: Determine the Phase Constant
The phase constant \( \phi \) is given as \( \frac{\pi}{3} \). This indicates the initial phase of the wave at \( t = 0 \) and \( x = 0 \).

### Conclusion
The expression represents a sinusoidal wave with:
- Amplitude \( A \),
- Wavelength \( \lambda = \frac{1}{5} \) m,
- Frequency \( f = 7.5 \) Hz,
- Phase constant \( \phi = \frac{\pi}{3} \).