Mathematics of Sound Objectives: Understand the concept of sound quality and what it represents. Describe the conditions which produce standing waves in a stretched string. Be able to describe the formation and characteristics of standing waves. Describe in qualitative terms the conditions for which interference of sound waves occur.
Let s Review What is Sound? The perception of a mechanical wave created by a vibration. Requires a medium for the wave to travel through. The medium the sound travel through does have an effect on the sound. It changes how fast the wave moves. Because the energy is transferred particle to particle, how those particles interact will determine how fast the energy can be passed along!
Transverse Velocity The two properties of a medium that determine how fast the sound will move is its elasticity and its inertia. On a string instrument, what can you change to change the sound you produce? Tightness of the string Thickness of the string For a wave in a string, these properties translate into Tension in the string Linear mass density (µ) of the string Mass/length Elastic property Inertial property F T - string tension m - string mass L - string length
Longitudinal Velocity Longitudinal waves have the same velocity controls. If the wave is traveling in a solid rod Elastic property --> Elastic Modulus (E) Inertial property --> Density (ρ) If the wave is traveling in a liquid or gas Elastic property --> Bulk Modulus (B) Inertial property --> Density (ρ) Elastic and Bulk Moduli have been determined for most materials!
Example! One end of a rope is tied to a stationary support at the top of a vertical mine shaft 80.0 m deep. A box of mineral samples with a mass of 20.0 kg is attached to the bottom of the rope. The mass of the rope is 2.00 kg. The geologist at the bottom of the mine signals to his colleague at the top by jerking the rope sideways. What type of wave is created? Transverse Wave Find: The wave s speed. The wavelength if at a point on the rope, the frequency is 2.00 Hz.
Solve It! What is the speed of the wave on the rope? F F T net = 0 F T - F g = 0 F g F T = F g = 20.0 kg 9.80 m/s 2 = 196 N 196 N 2.00 kg 80.0 m v = 88.5 m/s
Solve It Part II! If a point on the rope is given a frequency of 2.00 Hz, what is the wavelength of this wave? v =λƒ λ = v/ƒ = (88.5 m/s)/(2.00 Hz) λ = 44.3 m
Example! What is the wavelength of a 6,000 Hz sound wave traveling along an iron rod, if the density of iron is 7.8 x 10 3 kg/m 3 and its elastic modulus is 100 x 10 9 N/m 2? v = λƒ λ = v/ƒ (3581 m/s)/(6000 Hz) v = 3581 m/s λ = 0.60 m
Good Old Air The speed of sound in air is also dependent on the air temperature and pressure. The air temperature affects the elastic property Higher the temperature, the greater the particle interactions. The air pressure affects the inertial property Influences the mass density. At normal, atmospheric pressure, the speed of sound in air can be determined by: v = 331 m/s + (0.6 m/s C)T Where T is in C At T = 0 C, the speed of sound is 331 m/s!
Wave Behavior: Reflection Reflection occurs when a wave encounters an obstacle. The wave bounces off the obstacle and returns to the source. Speed = Distance/Time If we know how long it takes for the reflection to return, then we can figure out the distance! Distance = Speed x Time t total = 0.99 s Distance = 343 m/s x 0.495 s This is how echolocation and sonar work!
What Happens During Reflection? It will depend on the density of the materials that exist at the boundary. Let s look at the boundary between a less dense material and a more dense material. Less Dense More Dense The reflected wave is inverted! As the last particle pulls up on the fixed point, the fixed point pulls back with an equal and opposite force. Since the more dense material is more difficult to move, the less dense material is pulled downward.
The Other Option A boundary exists between a more dense material and a less dense material. More Dense Less Dense As the last particle pulls up on the string, the less dense string will move with the displacement. Therefore, there is no force pulling downward. The reflected wave is NOT inverted!
What s Going To Happen? The two waves with different amplitudes shown are traveling toward each other in the same medium. What happens when they meet? A. They reflect off each and move in opposite directions. B. They stick together and move in the direction of the larger wave. C. They stick together and move in the direction of the smaller wave. D. They overlap for an instant and then continue on their way. Let s Find Out!
Wave Interference Situation when two waves traveling in the same medium meet at the same time. Medium experiences the net effect of the two individual waves. Principle of Superposition The resulting wave is equal to the algebraic sum of the two waves.
Where It All Comes Together! The interference only occurs when the waves overlap. The movement of each is unaltered by the interference. The waves continue to move through the medium after interference has occurred.
What Happens if What happens if both ends of a medium are fixed and you send a wave down that medium? What happens when it reaches the end? It is reflected --> Inverts What happens when it reaches the other end? It is reflected again! --> Inverts Assuming there is no energy lost in the reflection, this process will just keep going and going and going. What if the waves interfere? You get a standing wave.
Standing Wave Nodes: Points of destructive interference where the cord remains still. Antinodes: Points of constructive interference where the cord oscillates with maximum amplitude. Standing waves can occur at more than one frequency! These frequencies are known as the natural frequencies or resonant frequencies.
Fundamental Frequency - ƒ 1 L = 1 2 λ 1 When a cord oscillates at the fundamental frequency, the length of the cord is equal to half a wavelength! Antinodes = 1 Nodes = 2
Higher Than The Fundamental The other natural frequencies that can cause a standing wave are known as overtones. In the case of a string, these overtones are integral multiples of the fundamental frequency. f n =nf 1 Where f 1 is the fundamental frequency. Overtones are also referred to as harmonics. Since fundamental is the lowest frequency, it is often called the first harmonic. First overtone is the second harmonic Second overtone is the third harmonic Etc.
Second Harmonic/1st Overtone - ƒ 2 L = λ 2 When a cord oscillates at the second harmonic frequency, the length of the cord is equal to a full wavelength! Antinodes = 2 Nodes = 3
Third Harmonic/2nd Overtone - ƒ 3 L = 3 2 λ 3 When a cord oscillates at the third harmonic frequency, the length of the cord is equal to one and a half wavelengths! Antinodes = 3 Nodes = 4
Do You See A Pattern? Fundamental (f 1 ): 2nd Harmonic (f 2 ): 3rd Harmonic (f 3 ): L = 1 2 λ 1 L = 2 2 λ 2 L = 3 2 λ 3 What s the general formula? L = n 2 λ n Rearrange! λ n = 2 n L
What Does It Sound Like? If you hear the same note on a guitar and a flute, do they sound the same? Even though the loudness and pitch can be exactly the same, they have a different quality of sound. Described as Timbre or Tonal color Result of the Number of harmonics present Relative amplitude of the harmonic When a note is played, it is the superposition of the fundamental frequency plus all the additional harmonics. Since each instrument will have different combinations of harmonics with different amplitudes, they each have a distinct sound.
Middle C Violin Saxophone Flute Bassoon
Example! A sound with a fundamental frequency of 294 Hz is sounded by a vibrating string. The length of the string is 70.0 cm. What is the frequency of the first three overtones? What is the speed of the standing wave produced if it looks like the following? First overtone = 2 nd Harmonic f 2 = 2(294 Hz) =588 Hz Second overtone = 3 rd Harmonic f 3 = 3(294 Hz) =882 Hz Third overtone = 4 th Harmonic f 4 = 4(294 Hz) =1176Hz
Example! Info: f 1 = 294 Hz L =.70 m L = 3 2 λ 3 λ 3 = 2 3 L λ 3= 2 3 (.70m) = 0.47 m λ 3 = 0.47 m v = f 3 λ 3 = (882 Hz)(0.47 m) f 3 = 882 Hz v = 412 m/s
Another Thing To Consider! Why do I close the door when I teach? I = P 4πR 2 Sound moves out in all directions, and its intensity decreases with distance.
Let s Model It!
Diffraction What s Happening? The ability of a wave to change direction as they pass through an opening or around a barrier. The amount of diffraction is equal to the sharpness of the bend. What factors influence the amount of diffraction? Wavelength Size of the opening Distance between source and barrier