Page 230 - Physics Coursebook 2015 (A level)

Cambridge International AS Level Physics
BOX 15.3: Reducing and eliminating errors (continued)
218
Summary
■■ Stationary waves are formed when two identical waves travelling in opposite directions meet and superpose. This usually happens when one wave is a reflection of the other.
■■ A stationary wave has a characteristic pattern of nodes and antinodes.
■■ A node is a point where the amplitude is always zero.
■■ An antinode is a point of maximum amplitude.
■■ Adjacent nodes (or adjacent antinodes) are separated
by a distance equal to half a wavelength.
■■ We can use the wave equation v = f λ to determine the speed v or the frequency f of a progressive wave.
The wavelength λ is found using the nodes or antinodes of the stationary wave pattern.
The antinode is at a distance c beyond the end
of the tube, where c is called the end-correction. Unfortunately, we do not know the value of c. It cannot be measured directly. However, we can write:
for the shorter tube, 4λ = l1 + c for the longer tube, 3λ = l + c
So, although we do not know the value of c, we can make two measurements (l1 and l2) and obtain an accurate value of λ. (You may be able to see from Figure 15.18 that the difference in lengths of the two tubes is indeed equal to half a wavelength.)
The end-correction c is an example of a systematic error. When we measure the length l of the tube, we are measuring a length which is consistently less than the quantity we really need to know (l + c). However, by understanding how the systematic error affects the results, we have been able to remove it from our measurements.
You will find more detailed discussion of systematic errors in Chapter P1: Practical skills at AS level.
4 2
Subtracting the first equation from the second equation gives:
3λ λ
4 − 4 = (l2 + c) − (l1 + c)
Simplifying gives: 2λ = l 2 − l 1
Hence:
λ = 2(l2 − l1)
QUESTION
7 In a resonance tube experiment, resonance is obtained for sound waves of frequency 630 Hz when the length of the air column is 12.6 cm and again when it is 38.8 cm. Determine:
a the wavelength of the sound waves causing resonance
b the end-correction for this tube
c the speed of sound in air.