Lecture19 - Lecture 19 The Wave Equation Last time we...

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Last time we started using Newton’s Laws to understand how a wave moves on a string T T dx θ 1 θ 2 T is the tension in the string Lecture 19: The Wave Equation
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We found that: How do we solve this equation? by “solve” I mean we need to find a function y ( x,t ) for which the wave equation is true for any value of x and t The wave equation is a 2nd-order partial differential equation in general, such equations can be a challenge to solve you’ll probably take a whole course on it But luckily, this one isn’t so bad In fact, we already know the answer: T 2 ψ ξ 2 = μ 2 ψ τ 2 This is the wave equation Useful for any kind of wave y x , t ( 29 = Α σιν κξ - ϖτ ( 29
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To check if it works, compute the derivatives: and then plug them into the wave equation: y x = Ak cos kx - wt ( ) 2 y x 2 = - Ak 2 sin kx - wt ( ) y t = - Aw cos kx - wt ( ) 2 y t 2 = - Aw 2 sin kx - wt ( ) T 2 ψ ξ 2 = μ 2 ψ τ 2 Τ - Ακ 2 σιν κξ - ϖτ ( 29 = μ
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