Application of Laplace Wave Equation in Music

Luckshay Batra

Table of Contents

• Laplace Equation• Problem Description• Motion of Stretched String• Solution• Conclusion• References

Laplace Equation

Partial Differential Equation does not fully simulate real life problems, one of them is that of Wave propagation.

Laplace Equation is the equation which is quite helpful in describing this phenomena.

I will present how this is applied to Music.

Problem Description Derive an equation governing

small transverse vibrations of an elastic string which is stretched of length “l” and then fixed at end points.

Let string be distorted and let at time t=0 , it be released and allowed to vibrate.

The Problem is to obtain the deflection y(x , t) at point x and at any time t>0

To solve this equation we have One Dimensional Wave equation.

Transverse Vibration of a Stretched String

Solution

• Solution to above equation is

Conclusion

The wave equation and its variants are present in every aspect of sound wave production and propagation.

The proper design of musical instruments, concert halls, and, for that matter, any room or device intended to produce or absorb sound all depend on an understanding of the principles behind and resulting from the wave equation.

References

• Main, Iain G. Vibrations and Waves in Physics. Cambridge: Cambridge UP, 1993. 3rd ed. Newton, Isaac.

• The Mathematical Principles of Natural Philosophy. Trans. Andrew Motte.

• London: Dawsons of Pall Mall, 1968. v. 2. Pierce, Allan D. Acoustics: An Introduction to Its Physical Principles and Applications. New York: McGraw-Hill, 1981.

• Book : Advanced Engineering Mathematics , by Dr. A.B. Mathur , V.P. Jaggi