derivation of wave equation
Let a string of matter be tightened between the points and of the -axis and let the string be made vibrate in the -plane. Let the of the string be the constant . We suppose that the amplitude of the vibration is so small that the tension of the string can be regarded to be constant.
The position of the string may be represented as a function
where is the time. We consider an element of the string situated on a tiny interval ; thus its mass is . If the angles the vector at the ends and of the element forms with the direction of the -axis are and , then the scalar force of all on (the gravitation omitted) are
Since the angles and are very small, the ratio
having the expression , also is very small. Therefore we can omit the horizontal component and think that the vibration of all elements is strictly vertical. Because of the smallness of the angles and , their sines in the expression of may be replaced with their tangents, and accordingly
the last form due to the mean-value theorem.
On the other hand, by Newton the force equals the mass times the acceleration:
Equating both expressions, dividing by and denoting , we obtain the partial differential equation
for the equation of the vibrating string.
But the equation (1) don’t suffice to entirely determine the vibration. Since the end of the string are immovable,the function has in to satisfy the boundary conditions
The vibration becomes completely determined when we know still e.g. at the beginning the position of the string and the initial velocity of the points of the string; so there should be the initial conditions
The equation (1) is a special case of the general wave equation
where . The equation (4) rules the spatial waves in . The number can be shown to be the velocity of propagation of the wave motion.
- 1 K. Väisälä: Matematiikka IV. Handout Nr. 141. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).
|Title||derivation of wave equation|
|Date of creation||2013-03-22 18:46:36|
|Last modified on||2013-03-22 18:46:36|
|Last modified by||pahio (2872)|