A wave `y_i = 0.3 cos (2.0x - 40t)` is travelling along a string toward a boundary at x=0. Write expressions for the reflected waves if .
(a) the string has a fixed end at x=0 and
(b) The string has a free end at x=0.
Assume SI units.

Explanation

To solve the problem, we need to analyze the behavior of the wave when it encounters two different types of boundaries: a fixed end and a free end. The given incident wave is:

\[ y_i = 0.3 \cos(2.0x - 40t) \]

### Step 1: Identify the incident wave parameters
The wave function is given in the form of \( y_i = A \cos(kx - \omega t) \), where:
- Amplitude \( A = 0.3 \)
- Wave number \( k = 2.0 \) (which relates to wavelength)
- Angular frequency \( \omega = 40 \) (which relates to frequency)

### Step 2: Determine the reflected wave for a fixed end
At a fixed boundary, the wave reflects with a phase change of \( \pi \) (or 180 degrees). This means the reflected wave will have the same amplitude but will change the sign of the cosine function. Therefore, the reflected wave \( y_r \) can be expressed as:

\[ y_r = -A \cos(kx + \omega t) \]

Substituting the values of \( A \), \( k \), and \( \omega \):

\[ y_r = -0.3 \cos(2.0x + 40t) \]

### Step 3: Determine the reflected wave for a free end
At a free boundary, the wave reflects without any phase change. This means the reflected wave will have the same amplitude and will maintain the same cosine function but will change the direction of propagation. Thus, the reflected wave \( y_r \) is given by:

\[ y_r = A \cos(kx + \omega t) \]

Substituting the values of \( A \), \( k \), and \( \omega \):

\[ y_r = 0.3 \cos(2.0x + 40t) \]

### Final Expressions
- For the fixed end at \( x = 0 \):
\[ y_r = -0.3 \cos(2.0x + 40t) \]

- For the free end at \( x = 0 \):
\[ y_r = 0.3 \cos(2.0x + 40t) \]

### Summary of Results
1. **Fixed End Reflection**: \( y_r = -0.3 \cos(2.0x + 40t) \)
2. **Free End Reflection**: \( y_r = 0.3 \cos(2.0x + 40t) \)

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