PHYS 2425 – Engineering Physics I

Standing Waves

 

Leader: _________________________          Recorder: __________________________

Skeptic: _________________________         Encourager: ________________________

 

Materials

Slinky

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

 

Introduction

      In this lab we will investigate the properties of standing waves. A standing wave is a wave that oscillates in time only but does not propagate like a traveling wave does.  The standing wave oscillates back and forth between two extremes called the envelope of the standing wave.   Standing waves are produced when a medium is subjected to boundary conditions.  A boundary condition forces the medium to move in a certain way at the boundary.  For instance, in a fixed boundary condition, the wave is not allowed to move at the boundary.  Mathematically if the amplitude of the wave is given by y(x,t), then the boundary condition that there is a fixed boundary at x = L can be expressed as y(L,t) = 0.  If we hold a flexible sting or a slinky firmly at both ends, then we have fixed boundaries at both ends of the slinky.   A standing wave can be thought as the resulting wave from wave interference.  Specifically if harmonic waves of equal amplitude travel to the left and right on the medium a standing wave results.

 

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

 

 

 

 

 

 

 

 

 

 

 


Q3)  Draw sketches of the envelopes of the next three standing wave patterns that satisfy the boundary conditions that the wave is fixed at both ends.

 

 

 

 

 

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).

 

 

 

Q5)  How does the number of nodes change with each successive standing wave pattern.

 

 

 

Q6)  The wavelength will be different for each successive standing wave pattern.  Find an expression for the wavelength of each pattern in terms of the number of nodes, n and L.

 

 

 

Procedure

 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.