Waves at Media Boundaries-Lesson 4 PDF

Waves at Media Boundaries Lesson 4 Learning Goal: E2.7 analyse the conditions required to produce resonance in vibrating objects and/or in air columns (e.g., in a string instrument, a tuning fork, a wind instrument), and explain how resonance is used in a variety of situations (e.g., to pro...

Waves at Media Boundaries Lesson 4 Learning Goal: E2.7 analyse the conditions required to produce resonance in vibrating objects and/or in air columns (e.g., in a string instrument, a tuning fork, a wind instrument), and explain how resonance is used in a variety of situations (e.g., to produce different notes in musical instruments; to limit undesirable vibrations in suspension bridges; to design buildings so that they do not resonate at the frequencies produced by earthquakes) E3.2 explain the components of resonance, and identify the conditions required for resonance to occur in vibrating objects and in various media (e.g., with reference to a musical instrument, a child on a swing, the Tacoma Narrows Bridge) Wave speed Sound wave speed depends on some of the properties of the medium through which the wave travels. The speed of sound in air depends on air temperature. Behaviour of sound wave is not visible to analyze so ropes are used for visual understanding.  What happens to the sound wave when the medium changes? Wave is partly reflected and partly transmitted. When sound waves move from one medium to another, there will be changes to the velocity (or speed), frequency and wavelength of the sound wave. This change in velocity can also result in a change of direction of the sound wave - also known as refraction.  Fixed-End reflection Reflection and Transmission of Sound Waves when it travels form slower to faster medium Reflection of Sound wave when it travels form slower to faster medium (Fixed-End reflection): When wave travels from less dense(slower medium) to more dense(faster) medium, the wave moving toward the boundary will be reflected and the reflected pulse has the same shape as the incoming pulse, but its orientation is inverted. More dense – faster medium Less dense – slower medium Transmission of sound waves when it travels from Slower(less dense) to Faster(more dense) Medium When a wave moves into a faster medium, then the wave splits into two, and one wave is reflected and the other is transmitted. The reflected wave is inverted. Slower medium Faster medium  Free -End reflection Reflection and Transmission of Sound Waves when it travels form faster to slower medium Reflection of Sound wave when it travels form faster to slower medium (Free -End reflection): When wave travels from more dense(faster medium) to less dense(slower) medium, the wave moving toward the boundary will be reflected in the same orientation as the incoming wave and with the same amplitude as the incoming wave. Faster(more dense) to slower(less dense) medium: If the wave moving along the rope encounters a medium that has a slower wave speed, then the wave splits into two, and one wave is reflected and the other is transmitted. The reflected wave is upright. faster medium slower medium What happens to the amplitude during reflection and transmission? The amplitude of the original wave may not be shared equally by the reflected wave and the transmitted wave. However, the sum of the two amplitudes must equal the amplitude of the original wave. What are standing waves A standing wave occurs when two waves of the same frequency and amplitude are moving in opposite directions and interfere with each other. It has certain points (called nodes) where the amplitude is always zero, and other points (called antinodes) where the amplitude fluctuates with maximum intensity. Node – zero amplitude Antinode – maximum amplitude Standing Waves between two fixed ends: Because it is difficult to draw a standing wave in motion, they are often illustrated showing both extremes at once. Standing waves in motion video https:// www.youtube.com/watch?v=no7ZPPqtZEg Harmonic s Standing waves and Harmonics All objects have a frequency or set of frequencies with which they naturally vibrate when struck, plucked, strummed or somehow disturbed. Each of the natural frequencies at which an object vibrates is associated with a standing wave pattern. The set of all possible standing waves are known as the harmonics of a system. Harmonics The simplest of the harmonics is called the fundamental or first harmonic. Subsequent standing waves are called the second harmonic, third harmonic, etc. Wavelengths of standing waves How many wavelengths are illustrated in the diagram below? Wavelengths of standing waves How many wavelengths are illustrated in the diagram below? 2 String Harmonics The 1st harmonic is called the fundamental frequency. In first harmonic half wavelength(l) fits in the length of the string(L). In second Harmonic one wavelength(l) fits in the length of the string(L) l = 2L l=L Wavelength of String Harmonics l =2L l=L l = (2/3)L Length of String Harmonics If l =2L, L1 = 1l/2 If l = L, L2 = 2l/2 If l = (2/3)L, L3 = 3l/2 Length of nth harmonic can be calculated using Ln = nl/2 Frequency of String Harmonics Using universal wave equation (v = fl), frequency can be calculated (f = v/l) f = v/2L as l =2L or f1 = 1v/2L f = v/L as l = L or f2= 2v/2L f = 3v/2L as l = 2L/3 or f3 = 3v/2L Frequency of nth harmonic can be calculated using fn = nv/2L Practice Question 1 A string resonates with a fundamental frequency of 512 Hz. The speed of sound in the string is 1750 m/s. What is the length of the string? Given: f = 512 Hz v = 1750 m/s Unknown: L=? Equation: v = fl Practice Question 2 A guitar string has a frequency of 256 Hz and a length of 49.1 cm. A guitarist reduces the string's length by 12.8 cm by pressing on the string. What is the new frequency? Given: For 1st length For 2nd length f = 256 Hz L = 49.1 cm = 0.491 m Unknown: f=? Equation: Note that reducing the length v = fl increased the fundamental l=2L frequency. Practice Question 3: 1. A 0.70-m long guitar string vibrates with standing waves. The first harmonic has a frequency of 261 Hz. A) Sketch the standing wave patterns for the first three harmonics. B) B) Calculate the wavelength of the first three harmonics. C) C) Calculate the frequency of the first three harmonics. Textbook Work: Page 426, # 1, 2, 3, 4 and 5