The Wave Equation: a derivation
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Post on 25-Dec-2015
DESCRIPTIONA derivation of the wave equation for waves on strings.
The wave equation: a derivation. We consider a string with mass per unit length kg m-1. 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). 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. The tension of the string acts along the tangent at any point. Therefore the resultant force parallel to the y axis is: If P and Q are very close together then sss is very small: Hence resultant force is: The mass of PQ is and its acceleration is : This could be written as: We have . Differentiating: So we have: Rearranging: Given that: we have: We know that: Therefore: and So we can write: If the amplitude of the oscillations is small enough: We put: So: The wave equation is usually written:
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