The green parrot is trying to
dislodge the other bird. How fast
will the wave he creates travel?
15.9 - Interactive problem: wave speed in a string
In this interactive problem, two strings are tied together with a knot and stretched
between two hooks. String 1, on the left, is twice as long as string 2, on the right.
Both strings have the same tension.
In the simulation, each string is plucked at its hook at the same instant. The
resulting wave pulses travel inward toward the knot. The wave pulse in string 1
starts at twice the distance from the knot as the wave pulse in string 2.
You want the wave pulses to meet at the knot at the same instant. To accomplish
this, set the linear density of each segment of string. When you increase a string
segment’s linear density in this simulation, the string gets thicker.
The minimum linear densities you can set are 0.010 kg/m. Set a convenient linear
density for string 1; your choice for this linear density determines the appropriate
linear density for string 2, which you must calculate.
Enter these values using the dials in the control panel and press GO to start the
wave pulses. If they do not meet at the knot at the same instant, the simulation will
pause when either of the pulses reaches the knot. Press RESET and enter different
values for the linear densities to try again.
15.10 - Mathematical description of a wave
When a mechanical wave travels through a medium, the particles in the medium
oscillate. Consider the diagram in Concept 1 showing a transverse periodic wave. The
particles of the string oscillate vertically and the wave moves horizontally.
The vertical displacement of the highlighted particle will change over time as it
oscillates. We show its displacement in Concept 1 at an instant in time.
In this section, we analyze a wave in which the particles oscillate in simple harmonic
motion. An equation that includes the sine function is used to describe a particle’s
displacement. The equation relates the vertical displacement of the particle to various
factors: the horizontal position of the particle, the elapsed time, and the wave’s
amplitude, frequency and wavelength. When all these factors are known, the vertical
position of a point can be determined at any time t.
Equation 1 describes a wave moving from left to right. The variable y in the equation is
the vertical displacement of a particle at a given horizontal position away from its
equilibrium position at a particular time. To use the equation, you must assume the
wave has traveled the length of the string, and the time t is some time after this has occurred.
In the equation, the variable x is the particle’s position along the x axis, which does not change for a given particle. The variable A is the
wave’s amplitude; the variable Ȝ is the wavelength; and the variable ƒ is the wave’s frequency.
The argument of the sine function is called the phase. As a wave sweeps past a particle located at a horizontal position x, the phase changes
linearly with respect to the elapsed time t.The phase is an angle measured in radians. The angle in the wave equation must be expressed in
Particles
Oscillate in simple harmonic motion