A travelling wave has the frequency `upsilon` and the particle displacement amplitude `A`. For the wave the particle velocity amplitude is ………………… and the particle acceleration amplitude is ……………………..

Explanation

To solve the problem, we need to find the particle velocity amplitude and the particle acceleration amplitude for a traveling wave characterized by a frequency \( \nu \) and a particle displacement amplitude \( A \).

### Step-by-Step Solution:

1. **Wave Equation**:
The general form of the wave equation for a simple harmonic wave is given by:
\[ y(x, t) = A \sin(\omega t + \phi) \]
where \( A \) is the particle displacement amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase constant.

2. **Relate Frequency and Angular Frequency**:
The angular frequency \( \omega \) is related to the frequency \( \nu \) by the equation:
\[ \omega = 2\pi\nu \]

3. **Find Particle Velocity**:
The particle velocity \( v \) is the time derivative of the displacement \( y \):
\[ v = \frac{dy}{dt} = \frac{d}{dt}(A \sin(\omega t + \phi)) \]
Using the chain rule, we differentiate:
\[ v = A \omega \cos(\omega t + \phi) \]
The amplitude of the particle velocity \( V \) is given by the coefficient of the cosine term:
\[ V = A\omega \]

4. **Substituting for Angular Frequency**:
Now, substitute \( \omega = 2\pi\nu \) into the expression for velocity amplitude:
\[ V = A(2\pi\nu) = 2\pi A \nu \]

5. **Find Particle Acceleration**:
The particle acceleration \( a \) is the time derivative of the velocity \( v \):
\[ a = \frac{dv}{dt} = \frac{d}{dt}(A \omega \cos(\omega t + \phi)) \]
Again, using the chain rule:
\[ a = -A \omega^2 \sin(\omega t + \phi) \]
The amplitude of the particle acceleration \( A_a \) is given by:
\[ A_a = A \omega^2 \]

6. **Substituting for Angular Frequency**:
Substitute \( \omega = 2\pi\nu \) into the expression for acceleration amplitude:
\[ A_a = A(2\pi\nu)^2 = A(4\pi^2\nu^2) = 4\pi^2 A \nu^2 \]

### Final Answers:
- The particle velocity amplitude is \( 2\pi A \nu \).
- The particle acceleration amplitude is \( 4\pi^2 A \nu^2 \).