The Wave Equation: a derivation

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A derivation of the wave equation for waves on strings.

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<ul><li><p>The wave equation: a derivation. We consider a string with mass per unit length kg m-1. </p><p>We consider transverse vibrations (diplacement y of each particle is perpendicular to the direction of propagation). We assume a constant tension F in the string (although in practice, the tension will change if the string extends). </p><p>We consider the motion of a small element of the string , between P and Q. When in equilibrium the string lies along the x axis and PQ is at P0Q0. We let the displacement of PQ from the x axis be y. </p><p>The tension of the string acts along the tangent at any point. Therefore the resultant force parallel to the y axis is: </p><p> If P and Q are very close together then sss is very small: </p><p> Hence resultant force is: </p><p>The mass of PQ is and its acceleration is : </p><p> This could be written as: </p></li><li><p>We have . </p><p>Differentiating: </p><p> So we have: </p><p> Rearranging: </p><p> Given that: </p><p> we have: </p><p> We know that: </p><p> Therefore: </p><p> and </p></li><li><p> So we can write: </p><p> If the amplitude of the oscillations is small enough: </p><p> We put: </p><p> So: </p><p> The wave equation is usually written: </p></li></ul>