PHYS 2425 – Engineering Physics I
Standing Waves
Leader:
_________________________ Recorder:
__________________________
Skeptic:
_________________________ Encourager:
________________________
Stop Watch
Mechanical Wave Driver with
elastic cord
Function Generator with power
cord
BNC cable with BNC-Banana
adapter
Table C-clamp with pulley
Mass hanger with additional
100 g
Q1) Let yr
= A sin(kx – ωt) and yl = A sin(kx + ωt).
Show that yr + yl
= 2Asin(kx)cos(ωt)
Note that not just any
standing wave will satisfy the boundary conditions.
Q2) Find a condition for the wavelength of the
standing wave such that it satisfies the boundary conditions y(0,t) = 0 and
y(L,t) = 0.
Figure
1 shows the envelopes of the first two standing wave patterns that satisfy the
boundary conditions given in Q2. The
envelope is the limit between which the standing wave oscillates.
Figure
1 The first two envelopes (boundaries)
between which standing wave patterns oscillate
Q4) A node is a point in the standing wave which
doesn’t move. The points at the
boundaries are nodes but we don’t usually count them. Label the five sketches of the standing wave
patterns with the number of nodes (not counting the boundaries).
Stretch the slinky to a length of 2 m for
short slinkies and 4 m for long ones.
Hold one end of the slinky fixed and shake the slinky until the first
pattern shown in figure 1 appears. Note
this will only happen at a specific frequency.
Q7) Determine the frequency at which the slinky is
being shaken. Accomplish this by timing
10 complete oscillations. The frequency
will be given by f = 10/t
t =
f =
10/t =
We
can determine the wavelength from the geometry of the standing wave
pattern. Notice that half of a
wavelength fits in the distance L.
Mathematically we can write this as l/2 = L.
Q8) What is the wavelength for this standing
wave?
l =
Record
your result in the table below.
Shake
the slinky so that you produce the second standing wave pattern shown in figure
2.
Q9) What is the wavelength for this standing
wave?
Determine
and record the frequency for this standing wave in the table below.
Shake
the slinky so that you produce the standing pattern with two nodes.
Q10) What is the wavelength for this standing
wave?
Determine
and record the frequency for this standing wave in the table below.
Trial Time for 10 oscillations (t) Frequency (f=10/t) Nodes (n) Wavelength (l)
1.
2.
3.
Q11) Did the wavelength of the waves increase or
decrease as you found standing wave patterns at higher frequencies?
Q12) What type of relationship have you seen
before that shows this trend.
We
will now use a different apparatus to examine this relationship more
accurately. Connect a cord to the
Mechanical Wave Driver (MWD), run the cord over a pulley and suspend a 50 g mass
hanger from the other end and add an additional 100 g. Position the MWD so that the point the cord
connects to it is approximately 1 m from the point the cord runs over the
pulley.
Q13) Record the length between the pulley and the
MWD.
Connect
the output of the function generator to the MWD. Place the multiplier switch in the 10
position and turn the amplitude all the way down (fully counter
clockwise). Turn the amplitude slowly up
until the MWD is oscillating about 1 cm up and down. Adjust the frequency of the oscillator until
you generate the first standing wave pattern shown in figure 1. Record the frequency of the standing wave and
the wavelength in the table below. Repeat for four more standing wave
patterns. Note you may need to change
the multiplier setting to find all five patterns.
Trial Frequency Nodes (n) Wavelength
(l)
1.
2.
3.
4.
5.
Q14) In question Q13) you suggested what type of
relationship may be demonstrated by the data.
What variable would you graph frequency vs. to demonstrate that type of
relationship?
Q15) Use either Excel or LoggerPro to construct a
graph of the frequency versus the reciprocal of the wavelength, f vs. 1/l. Add the best
fit line that goes through your data. Be
sure to include the equation of the line on your chart. Record the equation of the chart below. Print and attach the graph.
Q16) What are the units of the slope? What type of physical quantity has these
units?
Q17) Did the line pass through the origin or at
least very close?
Q18) Let
v be the speed of the wave, f be the frequency and λ the wavelength. Write an equation for the relationship shown
by your graph.
Q19) What is the speed of waves on the string?
Q20) While conducting this procedure you probably noticed
a pattern in the frequencies for the different standing waves. Describe the pattern.
Q21) Combine your results for Q6) and Q18) to find
an expression for the frequency of the standing wave with n nodes in terms of
v, L, and n.
Q22) Does the expression you found in Q21) agree
with the pattern you described in Q20).
Explain how.