Application of Partial differential Equation in Music

1. Application of Laplace Wave Equation in
Music
Luckshay Batra

2. Table of Contents
• Laplace Equation
• Problem Description
• Motion of Stretched String
• Solution
• Conclusion
• References

3. 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.

4. 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.

5. Transverse Vibration of a
Stretched String

6. Solution
• Solution to above equation is

7. 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.

8. 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