WOLFRAM|DEMONSTRATIONS PROJECT

The Vibrating String

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initial data
piecewise polynomial
piecewise linear
velocity
0.5
x interval
0
initial string
evolving string
left wave
right wave
run
display modes
The solutions of the wave equation
∂
t,t
u=
2
c
∂
x,x
u
represent the motion of an idealized string where
u=u(x,t)
represents the deflection of a string along the axis
x
at a time
t.
Here, such solutions are represented.
Using the locators, you can construct approximations to a polynomial of arbitrary degree or a piecewise continuous function on the interval 0 to π. The trigger starts the solution with no initial velocity and shows the evolution of the string as a function of time. This solution is built through the so-called d'Alembert solutions, which are a superposition of left and right traveling waves. The construction of these solutions can be explicitly demonstrated by only plotting right or left traveling waves (better seen on a larger
x
interval). The evolving string is the superposition of both waves. The time evolutions of the first three Fourier modes of the solutions are shown on the left of the plot.