The Wave Equation
I was wondering how you would derive the wave equation. What are the basic equations that I need in order to derive it. Thank you
Answer
The wave equation actually shows up in a lot of different contexts in physics as it is just a mathematical relationship. A few examples are a fluid like water or air, a string, and electric and magnetic fields. In each case, waves propagate through the medium.
In electromagnetism, you can derive the wave equation from Maxwell's equations. In a vacuum (meaning no charges or currents present), the equations in differential form reduce to:
∇·E = 0
∇·B = 0
∇×E = -dB/dt
∇×B = μ0 ɛ0 dE/dt
Taking the curl of the third equation, we get
∇x(∇×E) = -d(∇×B)/dt = - μ0 ɛ0 d^2E/dt^2
Using a vector identity for the left hand side, we find
∇x(∇×E) = ∇(∇·E) - ∇^2 E = - ∇^2 E
So we find that ∇^2 E = μ0 ɛ0 d^2E/dt^2, which is the wave equation where the wave speed is c = 1/sqrt(μ0 ɛ0).
This is just one example of how the wave equation is found in nature. It might be more detail than you wanted right now, but it should show you that whenever you can derive the wave equation in a medium, you will get waves that propagate through it! In this case, the waves are electromagnetic, like light for example.
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The wave equation actually shows up in a lot of different contexts in physics as it is just a mathematical relationship. A few examples are a fluid like water or air, a string, and electric and magnetic fields. In each case, waves propagate through the medium.
In electromagnetism, you can derive the wave equation from Maxwell's equations. In a vacuum (meaning no charges or currents present), the equations in differential form reduce to:
∇·E = 0
∇·B = 0
∇×E = -dB/dt
∇×B = μ0 ɛ0 dE/dt
Taking the curl of the third equation, we get
∇x(∇×E) = -d(∇×B)/dt = - μ0 ɛ0 d^2E/dt^2
Using a vector identity for the left hand side, we find
∇x(∇×E) = ∇(∇·E) - ∇^2 E = - ∇^2 E
So we find that ∇^2 E = μ0 ɛ0 d^2E/dt^2, which is the wave equation where the wave speed is c = 1/sqrt(μ0 ɛ0).
This is just one example of how the wave equation is found in nature. It might be more detail than you wanted right now, but it should show you that whenever you can derive the wave equation in a medium, you will get waves that propagate through it! In this case, the waves are electromagnetic, like light for example.