Velocity Potential function for the combination of an incident wave and reflected wave from an angled wall

For a combination of an incident wave and a reflected wave from an angled wall, we can describe the **velocity potential function** using concepts from fluid dynamics and potential theory. The velocity potential \(\Phi\) is a scalar function whose gradient gives the velocity field, particularly useful for describing irrotational, incompressible flows like those in water waves.

Let’s consider an incident wave traveling towards a wall at an angle \(\theta\), with a reflected wave bouncing back off the wall. Both waves can be modeled using harmonic functions since they describe periodic motion. The incident wave can be expressed as a plane wave, and the reflected wave will depend on the boundary conditions of the wall.

### 1. **Incident Wave**

Assume the incident wave is traveling in the positive x-direction at an angle \(\theta\) to the wall (which could be aligned with the y-axis). The velocity potential for an incident wave is given by:

\[
\Phi_{\text{incident}} = A e^{i(kx \cos \theta + ky \sin \theta - \omega t)}
\]

where:
- \(A\) is the amplitude,
- \(k\) is the wave number (related to wavelength by \(k = \frac{2\pi}{\lambda}\)),
- \(\omega\) is the angular frequency (\(\omega = 2\pi f\), where \(f\) is the wave frequency),
- \(t\) is time,
- \(\theta\) is the angle of incidence,
- \(x, y\) are spatial coordinates.

### 2. **Reflected Wave**

The reflected wave will have the same frequency and wave number as the incident wave but will propagate in a different direction, depending on the angle of reflection (equal to \(\theta\) for a perfectly reflective wall). The velocity potential for the reflected wave is:

\[
\Phi_{\text{reflected}} = A e^{i(kx \cos \theta - ky \sin \theta - \omega t)}
\]

Here, the term \(-ky \sin \theta\) indicates that the wave is reflected and traveling in the opposite direction along the y-axis.

### 3. **Total Velocity Potential**

The total velocity potential \(\Phi_{\text{total}}\) is the sum of the incident and reflected waves:

\[
\Phi_{\text{total}} = \Phi_{\text{incident}} + \Phi_{\text{reflected}}
\]

Substituting the individual terms:

\[
\Phi_{\text{total}} = A e^{i(kx \cos \theta + ky \sin \theta - \omega t)} + A e^{i(kx \cos \theta - ky \sin \theta - \omega t)}
\]

Factoring the common terms:

\[
\Phi_{\text{total}} = A e^{i(kx \cos \theta - \omega t)} \left( e^{i ky \sin \theta} + e^{-i ky \sin \theta} \right)
\]

Using Euler’s formula:

\[
\Phi_{\text{total}} = 2A e^{i(kx \cos \theta - \omega t)} \cos(k y \sin \theta)
\]

Thus, the total velocity potential function is:

\[
\boxed{\Phi_{\text{total}} = 2A e^{i(kx \cos \theta - \omega t)} \cos(k y \sin \theta)}
\]

### Interpretation:
- The first exponential term \( e^{i(kx \cos \theta - \omega t)} \) represents the wave propagation along the x-direction.
- The cosine term \( \cos(k y \sin \theta) \) modulates the wave as a function of the distance from the wall along the y-direction.

This potential function describes the superposition of the incident and reflected waves, forming a standing wave pattern near the wall.

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Would you like further clarification or details about any part of this explanation? Here are 5 questions to deepen your understanding:

1. How does the velocity potential function change for a perfectly absorbing wall?
2. What are the boundary conditions applied at the wall to derive the reflected wave?
3. Can we extend this model to waves reflecting from curved surfaces?
4. How would the velocity potential function look if the wall was moving?
5. How is the velocity potential related to the physical velocity field of the wave?

**Tip:** The gradient of the velocity potential \(\nabla \Phi\) gives the velocity field, making it a powerful tool for analyzing fluid flow patterns.

Explore how to derive the velocity potential function for a combination of incident and reflected waves from an angled wall. This solution covers key fluid dynamics concepts such as wave propagation, standing waves, and the superposition of incident and reflected waves using Euler’s formula. Suitable for undergraduate students.

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