Waves and Oscilla-ons Lecture 12 – Standing Waves Textbook reference: 18.1-­‐18.4 Intro to Carl Sagan’s Cosmos series Last -me: Superposi-on, interference •  Speed of a wave on a string “T” is Tension now! v=
�
T
µ
As a wave moves from one medium to another: Velocity changes, frequency does NOT change, wavelength MUST change v = fλ
Superposi-on: Waves add algebraically. Interference occurs and depends on the phase difference φ. φ
φ
y1 + y2 = 2A sin(kx − ωt + ) cos( )
2
2
path difference
× 2π = phase difference
λ
Last -me: Superposi-on, interference Reflec%on: •  When a wave is reflected from a fixed end (more dense medium) it undergoes an inversion (phase change of π radians). •  When a wave is reflected from an end that is free to move (less dense medium) it does not undergo an inversion, there is no phase change. y1 = A sin (kx -­‐ ωt), y2 = A sin (kx -­‐ ωt + φ) y=2A sin (kx -­‐ ωt + φ/2) cos (φ/2) Summary of Interference •  Construc-ve interference occurs when φ = 0 –  Amplitude of the resultant is 2A •  Destruc-ve interference occurs when φ = nπ where n is an odd integer –  Amplitude is 0 •  General interference occurs when 0 < φ < 2π –  Amplitude is 0 < Aresultant < 2A Ques-on The lower path, r1, is fixed. The upper path, r2, can be varied by changing the height, h. Describe the rela-onship between the height and what you hear. Assume that ini-ally r1 = r2 and h=h0. h Standing Waves Ac.ve Figure 18.08 •  The diagrams above show standing-­‐wave paberns produced at various -mes by two waves of equal amplitude travelling in opposite direc%ons •  In a standing wave, the elements of the medium alternate between the extremes shown in (a) and (c) Standing Waves •  Assume two waves with the same amplitude, frequency and wavelength, travelling in opposite direc-ons in a medium –  y1 = A sin (kx – ωt) –  y2 = A sin (kx + ωt) •  They interfere according to the superposi-on principle. Standing Waves A+B
A−B
sin A + sin B = 2 sin(
) cos(
)
2
2
⇒ y1 + y2 = A sin(kx − ωt) + A sin(kx + ωt)
= 2A sin(kx) cos(ωt)
This is a standing wave. Standing Waves •  The resultant wave will be y = 2A sin(kx) cos(ωt)
•  This is the wave func-on of a standing wave –  There is no (kx – ωt) term, and therefore it is not a travelling wave •  In observing a standing wave, there is no sense of mo-on in the direc-on of propaga-on of either of the original waves 2A sin kx
Par-cle Mo-on in Standing Waves •  Every element in the medium oscillates in simple harmonic mo-on with the same frequency, ω •  However, the amplitude of the simple harmonic mo-on depends on the loca-on of the element within the medium, 2A sin kx
Homework Set 6: PHYS 1121: 3 PHYS 1131: 3 Defini-ons Nodes: points on a standing wave with zero amplitude. 2A sin kx = 0
⇒ kx = 0, π, 2π, 3π....
2π
k=
λ
λ
3λ
x = 0, , λ, , ...
2
2
Defini-ons An%nodes: points on a standing wave at which maximum displacement occurs sin(kx) = ±1
π 3π 5π 7π
⇒ kx = ,
,
,
, ...
2 2 2 2
λ 3λ 5λ 7λ
nλ
x = , , , , ... =
, n = 1, 3, 5, 7, 9...
4 4 4 4
4
Amplitudes relevant to describing wave mo-on •  Three types of amplitudes exist: –  The amplitude of the individual waves, A –  The amplitude of the simple harmonic mo-on of the elements in the medium, 2A sin kx –  The amplitude of the standing wave, 2A •  A given element in a standing wave vibrates within the constraints of the envelope func-on 2Asin kx, where x is the posi-on of the element in the medium Ques-on Two waves traveling in opposite direc-ons produce a standing wave. The individual wave func-ons are: y1 = 4.0 sin(3.0x − 2.0t)
y2 = 4.0 sin(3.0x + 2.0t)
Where x and y are measured in cen-meters and t is in seconds. (a)  Find the amplitude of the simple harmonic mo-on of the element of the medium located at x = 2.3 cm. (b)  Find the posi-on of the nodes and an-nodes if one end of the string is at x = 0. Let’s take a short break Boundary Condi-ons Must be nodes here What are the possibili-es? λ
L=
2
L=λ
Fundamental or First Harmonic Second Harmonic 3λ
L=
2
Third Harmonic Wavelengths and Frequencies 2L
λn =
, n = 1, 2, 3, ...
n
�
v
v
n
T
fn =
=n
=
λn
2L
2L µ
Fundamental frequency is when n = 1. The frequencies of the other allowed modes are integer mul-ples of the fundamental frequency. When a standing wave is set up on a string fixed at both ends, which of the following statements is true? 1.
2.
3.
4.
The number of nodes is equal to the number of an-nodes. The wavelength is equal to the length of the string divided by an integer. The frequency is equal to the number of nodes -mes the fundamental frequency. The shape of the string at any instant shows a symmetry about the midpoint of the string. When a standing wave is set up on a string fixed at both ends, which of the following statements is true? λn = 2L / n n = 1, 2, 3 1.
2.
3.
The number of nodes is equal to the number of an-nodes. The wavelength is equal to the length of the string divided by an integer. The frequency is equal to the number of nodes -mes the fundamental frequency. 4.  The shape of the string at any instant shows a symmetry about the midpoint of the string. 4. Choice 1 is incorrect because the number of nodes is one greater than the number of antinodes.
Choice 2 is only true for half of the modes; it is not true for any odd-numbered mode. Choice 3 would be
correct if we replace the word nodes with antinodes (exercise for the student……).
Standing Waves in a String: I Ac.ve Figure 18.10 Consider a string fixed at both ends The string has length L Standing waves are set up by a con-nuous superposi-on of waves incident on and reflected from the ends •  The ends of the strings must necessarily be nodes •
•
•
•
–  They are fixed and therefore must have zero displacement •  These boundary condi-on result in a set of normal modes of vibra-on –  Each mode has a characteris-c frequency Defini-ons Normal modes: describes the way in which a string fixed at both ends can vibrate, the fundamental mode, second harmonic, third harmonic etc. Quan%za%on: When only certain frequencies of oscilla-ons are allowed we say a system is quan-zed. Standing Waves in a String: II •  This is the first normal mode that is consistent with the boundary condi-ons •  There are nodes at both ends •  There is one an-node in the middle •  This is the longest wavelength mode –  ½λ = L so λ = 2L Standing Waves in String: III •  For consecu-ve normal modes add an an-node at each step –  i.e. second mode (c) corresponds to to λ = L –  i.e. third mode (d) corresponds to λ = 2L/3 Ruben’s Tube with MOAR FIRE When you have enough gas pressure in the tube, it is more impressive Standing Waves on String: IV •  The wavelengths of the normal modes for a string of length L fixed at both ends are λn = 2L / n n = 1, 2, 3, … –  n is the nth normal mode of oscilla-on •  Since v=fλ, the natural frequencies are v
n T
fn = n
=
= nf1
2L 2L µ
Ques-on The middle C string on a piano has a fundamental frequency of 262 Hz, and the string for the first A above middle C has a fundamental frequency of 440 Hz. (a) Calculate the frequencies of the next two harmonics on the C string. (b) If the A and C strings have the same linear mass density μ and length L, determine the ra-o of tensions in the two strings. Ques-on One end of a horizontal string is abached to a vibra-ng blade, and the other end passes over a pulley. A sphere of mass 2.00 kg hangs on the end of the string. The string is vibra-ng in its second harmonic. A container of water is raised under the sphere so that the sphere is completely submerged. In this configura-on it vibrates in its fiqh harmonic. What is the radius of the sphere? Ques-on Two waves simultaneously present on a long string have a phase difference ϕ between them so that a standing wave formed between their combina-on is described by φ
φ
y(x, t) = 2A sin(kx + ) cos(ωt + )
2
2
(a)  Derive this expression. (b)  Despite the presence of the phase angle ϕ, is it s-ll true that the nodes are one-­‐half wavelength apart? Explain. (c)  Are the nodes different in any way to how they would be if ϕ were zero? Explain. Ques-on A standing-­‐wave pabern is observed in a thin wire with a length 3.00 m. The wave func-on is y = 0.00200 sin(πx) cos(100πt)
Where x and y are in meters and t is in seconds. (a) How many loops does this pabern exhibit? (b) What is the fundamental frequency of vibra-on of the wire? (c) If the original frequency is held constant and the tension in the wire is increased by a factor of 9, how many loops are present in the new pabern? Ques-on High-­‐frequency sound can be used to produce standing-­‐wave vibra-ons in a wine glass. A standing-­‐
wave vibra-on in a wine glass is observed to have four nodes and four an-nodes equally spaced around the 20.0 cm circumference of the rim of the glass. If transverse waves move around the rim of the glass at 900 m/s, an opera singer would have to produce a high harmonic with what frequency to shaber the glass with a resonant vibra-on as shown in the figure.