Problem 1 - Classical Mechanics

Question

Show that under the Galilean transformation $$ \bar{x}=x, \quad \bar{y}=y, \quad \bar{z}=z-v t, \quad \bar{t}=t $$ the equation $$ \nabla^{2} \phi-\frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial t^{2}}=0 $$ becomes $$ \frac{\partial^{2} \phi}{\partial \bar{x}^{2}}+\frac{\partial^{2} \phi}{\partial \bar{y}^{2}}+\left(1-\frac{v^{2}}{c^{2}}\right) \frac{\partial^{2} \phi}{\partial \bar{z}^{2}}+\frac{2 v}{c^{2}} \frac{\partial^{2} \phi}{\partial \bar{t} \partial \bar{z}}-\frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial \bar{t}^{2}}=0 $$

Step-by-Step Solution

Answer

The wave equation transforms as described under the Galilean transformation.

Step 1: Understand the Galilean Transformation

The Galilean transformation is given by the mappings: \(\bar{x} = x\), \(\bar{y} = y\), \(\bar{z} = z - vt\), and \(\bar{t} = t\). These transformations affect the \(z\)-coordinate due to the presence of velocity \(v\). The unchanged \(t\) coordinate implies no time dilation or contraction.

Step 2: Apply Transformation to Spatial Derivatives

Under the transformation, the spatial derivative with respect to \(z\) transforms as follows: \(\frac{\partial}{\partial z} = \frac{\partial}{\partial \bar{z}} \) since \(\bar{z} = z - vt\). Thus, \(\frac{\partial^2}{\partial z^2} = \frac{\partial^2}{\partial \bar{z}^2}\). The partial derivatives in \(x\) and \(y\) remain unchanged as \(\frac{\partial^2}{\partial x^2} = \frac{\partial^2}{\partial \bar{x}^2}\) and \(\frac{\partial^2}{\partial y^2} = \frac{\partial^2}{\partial \bar{y}^2}\).

Step 3: Apply Transformation to Time Derivative

For a function \(\phi(x, y, z, t)\), under the transformation, the total time derivative changes, due to velocity \(v\): \(\frac{\partial}{\partial t} = \frac{\partial}{\partial \bar{t}} - v \frac{\partial}{\partial \bar{z}}\). Consequently, the second time derivative transforms as: \(\frac{\partial^2}{\partial t^2} = \frac{\partial^2}{\partial \bar{t}^2} - 2v \frac{\partial^2}{\partial \bar{t} \partial \bar{z}} + v^2 \frac{\partial^2}{\partial \bar{z}^2}\).

Step 4: Substitute into the Original Equation

Substitute the expressions for the derivatives from steps 2 and 3 into the original wave equation \(abla^2\phi - \frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2} = 0\). For \(abla^2\), substitute the spatial derivatives: \(abla^2\phi = \frac{\partial^2\phi}{\partial \bar{x}^2} + \frac{\partial^2\phi}{\partial \bar{y}^2} + \frac{\partial^2\phi}{\partial \bar{z}^2}\). Substitute the time derivative transformations: \(\frac{1}{c^2}\left(\frac{\partial^2\phi}{\partial \bar{t}^2} - 2v \frac{\partial^2\phi}{\partial \bar{t} \partial \bar{z}} + v^2 \frac{\partial^2\phi}{\partial \bar{z}^2}\right)\).

Step 5: Simplify and Rearrange the Transformed Equation

Reorganize the resulting equation from Step 4: Combine like terms to get: \(\frac{\partial^2 \phi}{\partial \bar{x}^{2}} + \frac{\partial^2 \phi}{\partial \bar{y}^{2}} + \left(1-\frac{v^{2}}{c^{2}}\right)\frac{\partial^2 \phi}{\partial \bar{z}^{2}} + \frac{2 v}{c^{2}}\frac{\partial^{2} \phi}{\partial \bar{t} \partial \bar{z}} - \frac{1}{c^{2}}\frac{\partial^{2} \phi}{\partial \bar{t}^{2}} = 0\). This matches the target transformed equation.

Key Concepts

Wave Equation

The wave equation is a critical component in physics and mathematics, describing how waveforms (such as sound or light waves) propagate through space over time. Mathematically, it is expressed as a second-order partial differential equation. In a simplified form, the wave equation is written as \( abla^2 \phi - \frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2} = 0 \), where \(\phi\) represents the wave function.

- \( abla^2 \phi \) or the Laplacian of \( \phi \) accounts for the spatial variation of the wave.
- The term \(-\frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2}\) depicts the change in the wave over time, with \(c\) as the speed of the wave in the medium.

The wave equation is foundational for understanding phenomena such as electromagnetic waves, sound waves, and water waves. In more advanced contexts, it assists in studying the behavior of materials under dynamic stress.

Coordinate Transformation

Coordinate transformations are essential for transitioning between different frames of reference, often facilitating problem-solving in physics. The Galilean transformation is a linear transformation used when examining a moving frame of reference at non-relativistic speeds.

Under a Galilean transformation:

• \( \bar{x} = x \)
• \( \bar{y} = y \)
• \( \bar{z} = z - vt \)
• \( \bar{t} = t \)

Here, \(v\) is the velocity of the moving frame relative to the original frame. This transformation simplifies many physical analyses by confining changes to the space domain (particularly the \(z\) direction) while keeping time constant.

Understanding coordinate transformations, like Galilean, allows for more accessible adaptation of solutions to various physical problems, such as transforming wave equations to reflect different observational frames.

Partial Derivatives

Partial derivatives are a cornerstone concept in calculus, especially in dealing with functions of multiple variables. They measure how a function changes as one specific variable is varied, while others remain constant.

- In the context of the wave equation, partial derivatives are crucial for breaking down how the wave function \(\phi\) changes with respect to each spatial coordinate \(x, y, z\) and time \(t\).

For example:

• The partial derivative \(\frac{\partial \phi}{\partial z}\) suggests how \(\phi\) varies along the \(z\)-axis alone.
• The second partial derivative \(\frac{\partial^2 \phi}{\partial t^2}\) captures the acceleration or the rate of change of change of the wave function with respect to time.

In transformations, partial derivatives are recalculated to accommodate shifts in the coordinate frame, like in the Galilean transformation. By understanding partial derivatives, one can dissect complex equations and interpret how multiple variables influence a system, making them vital in fields ranging from physics to engineering.