Reflexión de formadores en DIT sobre la situación final

A.2.1.6 A wave starting from one end reflects from the other end either in phase (open end) or Q degrees out of phase (closed end). The

two waves then interfere with each other such that at points of nodes the net displacement of air molecules is zero and at points of antinodes, the displacement is most. Maximum amplitude, i.e. antinodes occur when the two waves are in phase and minimal (or zero) amplitude, i.e. nodes occur when the two waves are completely out of phase. The amplitude of the antinodal positions vary from zero to maximum with time.

A.2.1.7 When the forward and reflected waves add constructively to create a displacement antinode, the air molecules at that point are

able to move the most hence creating a large amplitude. When the forward and reflected waves interfere destructively to form a node, the air molecules are compressed together and have the least amplitude of motion, hence have high pressure (pressure antinode) and zero displacement (displacement node).

A.2.3.1

Statement True or _false?

(a) In the fundamental standing wave, there is only one displacement antinode but two nodes. T (b) Standing waves are created by two waves of the same frequency and wavelength travelling in opposite directions. T (c) The fundamental standing wave is the loudest harmonic heard because the string vibrates with the highest amplitude. T (d) Only the first two harmonics correspond to resonant frequencies; the other harmonics do not. F (e) For a string of length L, the nth resonant frequency is given byf nv

L

n= ₂ , where v is the speed of sound. T

(f) For a string of length L, the nth resonant frequency is given by f nv L

n= ₄ , where v is the speed of sound. F

(g) The third harmonic is a frequency three times that of the fundamental. T (h) The third harmonic is a frequency two times that of the fundamental. F (i) All harmonics, not just the even ones, are possible in a guitar string. T (j) Only the odd number harmonics are possible in a guitar string. F (k) A standing wave in a guitar string is set up due to the reflection from fixed ends, and this reflection is M/2 out of phase

with the incident wave. T (l) The reflection from fixed ends in a guitar string are in phase with the incident wave thereby reinforcing the incident

wave and causing resonance. F (m) In the fundamental standing wave, there is only one pressure antinode but two nodes. F

A.2.3.2 (a) Soundwaves are reflected from open ends with a phase change of half a wavelength.

(b) At the ends of the flute there is a destructive interference and pressure nodes.

(c) The pressure of air is maximum in the middle of the flute and the pressure at the ends is equal to air pressure. (d) The displacement of air molecules is maximum at the ends of the flute and for the fundamental, the displacement of air

molecules is minimum in the middle of the flute.

A.2.3.3

Statement True or _false?

(a) The harmonics are very similar to that of a string tied at both ends. T (b) The harmonics are very similar to that of a string tied at one end and held loosely at the other end. F (c) The motion of air particles is the least where there is a pressure antinode and most where there is a pressure node. T (d) The motion of air particles is the least where there is a pressure node and most where there is a pressure antonode. F (e) The frequency of the third harmonic is three times that of the fundamental. T (f) The frequency of the third harmonic is 2/3 times that of the second harmonic. T

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A.2.3.4 (a) Displacement modes. (b) Pressure modes. Fundamental 2nd harmonic 3rd harmonic Fundamental 2nd harmonic 3rd harmonic

A.2.3.5 (a) Soundwaves are reflected from the closed end of the air column with a change of phase = nil. Therefore there is

constructive interference at the closed end and a pressure node results.

(b) The fundamental mode of vibration will have the relationship L =M

4 where L is the length of the air column. Only odd

number of harmonics are possible.

A.2.3.6 An air column above water is like a pipe closed at one end. The resonant frequencies follow the relationship f_n = nv/4L,

where n = 1, 3, 5, 7... As water is filled in, the length of air column, L, reduces thereby increasing the frequency.

A.2.3.7 (a) String fixed at both ends:

It can be shown that the harmonics are L L L L n f nv L n n 1 ₂ 2 3 2 2 3 2 2 2 = ,M = M, = M, ...  M corresponding to = .. (b) Pipe closed at one end: It can be shown that the harmonics are L1 ₄ L3 L5

3 4

5 4 = ,M = M, = M ... Only the odd harmonics are possible.

Thus, Ln=n

M

4, where n is an odd number. The relationship corresponds to f nv

L

n=₄ , where n is an odd number. (c) Pipe open at both ends: Same as string fixed at both ends.

A.2.3.8 Using f nv

L L n

n=₂ and n= ₂

M _{for pipes open at both ends and for strings, and f} nv L L

n

n=₄ and n= ₄

M _{for pipes closed at one end, the}

following results are obtained.

Ratio of: Ratio is:

Fundamental frequency to the 3rd harmonic for a string 1:3 2nd harmonic frequency to the 4th harmonic for a string 1:2 2nd harmonic frequency to the 3rd harmonic for a pipe open at both ends 2:3 The wavelength of the fundamental to the 3rd harmonic for a string 3:1 Fundamental frequency to the next higher frequency for a pipe closed at one end 1:3 The wavelength of the 2nd harmonic to the 4th harmonic for a pipe open at both ends 2:1

A.2.3.9 L v f = = = . 4 815 4 312× 0 65 m

A.2.3.10 End correction is 0.4d = 0.4 × 0.2 = 0.08 m. We would need to add this value to the calculated wavelength.

A.2.4.1 The energy of the oscillator is transmitted from one end of the medium through to the other in a travelling wave. However, this

energy changes the amplitude of the standing wave and is not transmitted from one end to the other in a standing wave.

A.2.4.2 Amplitude is the same for all string elements in a travelling sinusoidal wave. In a standing wave, amplitude varies with position

along the string. The positions of node have zero amplitude while positions of antinode have amplitude which oscillates from zero to maximum.

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A.2.4.3 A travelling (sinusoidal) wave of uniform speed can be represented as a graph of amplitude versus distance or amplitude versus