A travelling sound wave is described by the equation `y = 2sin(4t - 5x)` where `y` is measured in centimeter. `t` in seconds and `x` in meters.
(a) Find the ratio of amplitude and wavelength of wave.
(b) Find the ratio of maximum velocity of particular to wave velocity.

Explanation

To solve the problem, we will break it down into two parts as specified in the question.

### Given:
The equation of the traveling sound wave is:
\[ y = 2 \sin(4t - 5x) \]
where:
- \( y \) is in centimeters,
- \( t \) is in seconds,
- \( x \) is in meters.

### Part (a): Find the ratio of amplitude and wavelength of the wave.

1. **Identify the Amplitude**:
The amplitude \( A \) is the coefficient of the sine function in the wave equation. From the equation:
\[ A = 2 \, \text{cm} \]
To convert this to meters:
\[ A = 2 \times 10^{-2} \, \text{m} \]

2. **Identify the Angular Frequency and Wave Number**:
The angular frequency \( \omega \) and wave number \( k \) can be identified from the equation:
\[ \omega = 4 \, \text{rad/s} \]
\[ k = 5 \, \text{m}^{-1} \]

3. **Calculate the Wavelength**:
The wavelength \( \lambda \) is related to the wave number \( k \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Rearranging gives:
\[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{5} \]

4. **Calculate the Ratio of Amplitude to Wavelength**:
Now, we can find the ratio \( \frac{A}{\lambda} \):
\[ \frac{A}{\lambda} = \frac{2 \times 10^{-2}}{\frac{2\pi}{5}} = \frac{2 \times 10^{-2} \times 5}{2\pi} = \frac{5 \times 10^{-2}}{\pi} \]

### Part (b): Find the ratio of maximum velocity to wave velocity.

1. **Calculate Maximum Velocity**:
The maximum velocity \( V_m \) of a particle in the wave is given by:
\[ V_m = A \cdot \omega \]
Substituting the values:
\[ V_m = (2 \times 10^{-2}) \cdot 4 = 8 \times 10^{-2} \, \text{m/s} \]

2. **Calculate Wave Velocity**:
The wave velocity \( v \) is given by:
\[ v = \frac{\omega}{k} \]
Substituting the values:
\[ v = \frac{4}{5} = 0.8 \, \text{m/s} \]

3. **Calculate the Ratio of Maximum Velocity to Wave Velocity**:
Now, we find the ratio \( \frac{V_m}{v} \):
\[ \frac{V_m}{v} = \frac{8 \times 10^{-2}}{0.8} = 0.1 \]

### Final Answers:
(a) The ratio of amplitude to wavelength is \( \frac{5 \times 10^{-2}}{\pi} \).

(b) The ratio of maximum velocity to wave velocity is \( 0.1 \).