derivation of wave equation
Let a string of homogeneous matter be tightened between the points
and
of the
-axis and let the string be made vibrate in the
-plane. Let the line density of mass 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
components of the resultant force
of all forces 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
 |
(1) |
for the equation of the transversely 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 addition to satisfy the boundary
conditions
 |
(2) |
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
 |
(3) |
The equation (1) is a special case of the general wave equation
 |
(4) |
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. V¨AISÄLÄ: Matematiikka IV. Handout Nr. 141. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).
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