STANDING WAVES

WHAT IS A STANDING WAVE? A standing wave is created when the waves from the source (my hand) interfere with the reflected waves in such a manner as to create points along the medium (our spring) that appear to be standing still. It looks like it is standing still but the waves are traveling through it. Standing waves are only created at specific frequencies, called harmonic frequencies.

DEFINITIONS Standing Wave : A pattern which results from the interference of identical waves traveling in opposite directions at a SPECIFIC frequency. The result is a wave that appears to be standing still. Node : A point of TDI, this point is never displaced. Antinode : A point along the medium which alternates between the maximum positive and the maximum negative displacement. Constructive interference takes place here resulting in large crests and troughs. These are the “loops.” Harmonics or Harmonic Frequencies : Specific frequencies of waves that create a standing wave pattern. Corresponds to the number of loops/antinodes in a standing wave.

COMPONENTS OF A STANDING WAVE 1 L = length of the medium The medium MUST have a fixed length 1 λ = 2 loops 1 2 3

COMPONENTS OF A STANDING WAVE On the 1 st diagram, how many wavelengths do you see? 1/2 λ How long is 1 λ? 1λ = 2 Length of String On the 2 nd diagram, how many wavelengths do you see? 1λ How long is 1λ 1λ = 1 Length of String On the 3 rd diagram, how long will the wavelength be? 1λ = 2/3 Length of String 1 2 3

PATTERN/FORMULA FOR WAVELENGTH IN STANDING WAVES If the # of harmonics are represented by the # of loops in the standing wave, the wavelength can be determined using the following formula: λ = (2L)/n {n=1,2,3,4...} Where “n” represents the harmonic number or number of loops “n” must be a whole number “L” represents the length of the medium 1 2 3

PATTERN/FORMULA FOR WAVELENGTH IN STANDING WAVES Notice in the 3 diagrams, this pattern is true. Let’s say 1L = 1 meter 1) λ = 2 (1m)/1 = 2 m 2) λ = 2 (1m)/2 = 1 m 3) λ = 2 (1m)/3 = 2/3 m 1 2 3

CONDITIONS FOR STANDING WAVES TO EXIST 1. There must be two waves of the same frequency. One or both must be a reflected wave. 2. The number of harmonics or “n” must be a whole number If not, a standing wave is not possible and will not form 3. There must either be a whole or a half number of wavelengths in the pattern. For example, 2λ or 1.5λ, but not 1.9λ or 3.25 λ. This is because the end points of the wave must be a node, which can not exist at any other lengths other than a multiple of ½ λ.

QUESTIONS WITH STANDING WAVES Example 1: How many nodes are there on this standing wave? (8) How many wavelengths are in this standing wave? (3.5) How many antinodes or loops? (7) What Harmonic is the wave in? (Seventh)

QUESTIONS WITH STANDING WAVES Example 2: Our spring is 9.0m long, and we create a fifth harmonic standing wave. What is the wavelength? L = 9m n=5 λ=? λ =2L/n λ =2(9.0m)/5 λ =18/5 λ = 3.6 m

QUESTIONS WITH STANDING WAVES Example 3: Our spring is 9.0 m long and our wavelength is 2.5 m, can we create a standing wave? L = 9.0m n=? λ=2.5 λ=2L/n n=2L/λ n=2(9.0m)/2.5m n= 18m/2.5m n= not a whole number, therefore, no standing wave possible

QUESTIONS WITH STANDING WAVES Example 4: The distance between two successive nodes in a vibrating spring is 14 cm. The frequency of the source is 35 Hz. A) What is the wavelength? Distance between 2 nodes is 1/2 (half of a wavelength). wavelength is 2(14 cm) = 28cm B) What is the velocity of the waves? v = ƒ v = (35Hz)(28cm) v = 9.8 x 10 2 cm/s