To solve the problem step by step, we will analyze the given wave equation and extract the required information.
**Given Wave Equation:**
\[ y(x,t) = 3.0 \sin(36t + 0.018x + \frac{\pi}{4}) \]
where \( x \) and \( y \) are in cm, and \( t \) is in seconds.
### Step 1: Determine if the wave is traveling or stationary
The general form of a traveling wave is:
\[ y(x,t) = A \sin(\omega t + kx + \phi) \]
Here, the term \( \omega t + kx \) indicates the wave's propagation direction. If it is of the form \( \omega t - kx \), the wave travels in the positive x-direction; if it is \( \omega t + kx \), it travels in the negative x-direction.
**Analysis:**
- In our equation, we have \( 36t + 0.018x \), which indicates that the wave is traveling in the negative x-direction.
### Step 2: Calculate the speed of the wave
The speed \( v \) of a wave can be calculated using the relationship:
\[ v = \frac{\omega}{k} \]
Where:
- \( \omega = 36 \, \text{rad/s} \)
- \( k = 0.018 \, \text{cm}^{-1} \)
**Calculation:**
1. Convert \( k \) to SI units:
\[ k = 0.018 \, \text{cm}^{-1} = 0.018 \times 100 \, \text{m}^{-1} = 1.8 \, \text{m}^{-1} \]
2. Calculate the speed:
\[ v = \frac{\omega}{k} = \frac{36 \, \text{rad/s}}{1.8 \, \text{m}^{-1}} = 20 \, \text{m/s} \]
### Step 3: Determine the amplitude and frequency
- The amplitude \( A \) is the coefficient in front of the sine function, which is \( 3.0 \, \text{cm} \).
- The frequency \( f \) can be calculated using:
\[ \omega = 2\pi f \]
Thus,
\[ f = \frac{\omega}{2\pi} = \frac{36}{2\pi} \approx \frac{36}{6.28} \approx 5.73 \, \text{Hz} \]
### Step 4: Find the initial phase at the origin
The initial phase at the origin (when \( x = 0 \) and \( t = 0 \)) can be found by substituting these values into the wave equation:
\[ y(0,0) = 3.0 \sin(0 + 0 + \frac{\pi}{4}) = 3.0 \sin(\frac{\pi}{4}) = 3.0 \cdot \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2} \, \text{cm} \]
The initial phase is \( \frac{\pi}{4} \).
### Step 5: Calculate the least distance between two successive crests (wavelength)
The wavelength \( \lambda \) can be found using:
\[ k = \frac{2\pi}{\lambda} \]
Thus,
\[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{0.018} \approx \frac{6.28}{0.018} \approx 348.89 \, \text{cm} \]
### Summary of Results:
1. The wave is traveling in the negative x-direction.
2. Speed of the wave: \( 20 \, \text{m/s} \).
3. Amplitude: \( 3.0 \, \text{cm} \).
4. Frequency: \( 5.7 \, \text{Hz} \).
5. Initial phase at the origin: \( \frac{\pi}{4} \).
6. Wavelength (least distance between two successive crests): \( 348.89 \, \text{cm} \).
