G11-6-1 Vibration of Air Columns (Grade 11) PDF

VIBRATION OF AIR COLUMNS Grade 11 Grade 1 Most musical instruments produce sound due to vibration of the string and air column. These vibrations give rise to waves known as stationary waves in the string (or) air column and cause progressive waves to spread out from musical instrument. Learning Outc...

VIBRATION OF AIR COLUMNS Grade 11 Grade 1 Most musical instruments produce sound due to vibration of the string and air column. These vibrations give rise to waves known as stationary waves in the string (or) air column and cause progressive waves to spread out from musical instrument. Learning Outcomes It is expected that students will ⚫ analyse the characteristics of stationary waves.. investigate vibrating strings. examine sound produced by resonance columns and organ pipes. ⚫ explain intensity of waves.. acquire basic knowledge of generation and propagation of waves. Nod- One are calle Waves are classified as progressive waves and stationary waves. Progressive waves spread out from the region in which they are produced. Unlike progressive waves, stationary waves do not spread out but remain in the region in which they are produced. So they are also called standing waves. Sound waves which travel in air when we speak and water waves which travel on the water surface when a stone is dropped are progressive waves. Progressive waves carry energy through the medium. The waves produced in hollow tubes such as flutes and in string instruments such as violins and mandolins are stationary waves. 6.1 STATIONARY WAVES A stationary wave is the resultant wave by the superposition of two waves of the same type having equal amplitudes and velocities traveling in opposite directions. The formation of a stationary wave can be demonstrated as follows. One end of a string is fastened to the vibrating arm of an electric vibrator and the other end is held by the hand. When the electric vibrator is activated while the string is held tightly, the string vibrates due to the electric vibrator. The incident wave travels from the vibrator to the hand and the reflected wave travels from the hand to the vibrator. The resultant wave obtained from the superposition of the incident and the reflected wave is a stationary wave as shown Figure 6.1. Principle of Superposition electric vibrator Figure 6.1 Illustration of production of stationary wave Waves have linearity property. When two (or) more waves pass the same point, the resultant wave at that point is the sum of the individual waves. This is called principle of superposition. If the resultant wave has a larger amplitude than that of individual waves, this is said to be constructive interference. If the resultant wave has a smaller amplitude than that of individual waves, this is said to be destructive interference. If the resultant wave has a zero amplitude, it is said to be completely destructive interference in Figure 6.2. The nod F is a 68 ric Nodes and Antinodes Figure 6.2 Superposition of two waves One characteristic of every stationary wave pattern is that there are points along the medium, which are called nodes and antinodes. The points marked N in Figure 6.3 are always stationary. They are called nodes. The points between nodes are vibrating with different amplitudes. The mid-points between successive nodes have the largest amplitudes and are called antinodes which are marked A in Figure 6.3. displacement (v) 0000 Figure 6.3 Illustration of nodes and antinodes distance (x) Nodes and antinodes always alternate and are equally spaced. The distance between two successive nodes (or) antinodes is equal to 2 where 2 is the wavelength. The distance from a node to the nearest antinodes is equal to 2. 4 2 Example 6.1 If the distance between two consecutive nodes of a stationary wave in a stretched string is 0.5 m, (i) find the distance between two successive antinodes, (ii) find the distance between a node and the nearest antinode. 2 distance between two consecutive nodes = 0.5 m, λ = 1 m 2 (i) distance between two successive antinodes 2-0.5 m 2 λ (ii) distance between a node and the nearest antinode -=0.25 m 4 ve at ve aid ely Reviewed Exercise 1. Describe how stationary waves can be produced. 2. How are antinodes and nodes created in a stationary wave? (Hints: Nodes/Antinodes are produced at the locations where destructive /constructive interference occurs.) Key Words: progressive wave, stationary wave, superposition, nodes, antinodes 69 6.2 VIBRATING STRINGS In most of the musical instruments (for example, violin, mandolin, etc.) the stretched strings act as source of sound. When the stretched strings are plucked, the stationary waves are produced. They have certain specific frequencies. To understand why only certain frequencies can occur, consider a of length / rigidly fixed at both ends. string When the string is plucked the stationary waves with nodes at the fixed ends are formed. The waves that are formed on the string are called harmonics. The first four harmonics of the vibrating string are shown in Figure 6.4. 7 1=2.3=21 First harmonic (fundamental) W F C A N 1=222, 22-23/2 21 = Second harmonic (first overtone) A IN 1=32, 2=21 23 Third harmonic (second overtone) 3 N 1=424, 7=24 21 =- Fourth harmonic (third overtone) 4 A A Figure 6.4 Harmonics of vibrating string The wavelength in Figure 6.4 can be labeled with a subscript n, where n is a positive integer and is called harmonic number. For the nth harmonic 21 2 = n (n = 1, 2, 3,....) (6.1) The corresponding frequencies are calculated from v = f, where v is the velocity of wave in a string. The frequencies of vibrating string of length / are, την fn= 21 (n = 1, 2, 3,....) (6.2) Vibration of a string in one single segment is called the fundamental (or) first harmonic. A musical tone which is part of the harmonic series above a fundamental note is called an overtone The velocity of a wave in a vibrating string depends on the tension of the string and mass per length of the string as follows. unit V=√ where T = tension of the string μ = mass per unit length of the string The frequencies of the vibrating string can also be written in terms of T and μ. 70 (6.3) We obtain (n = 1, 2, 3,...) (6.4) e μ g T First harmonic, n = 1, fi 21 \ μ S 2T e Second harmonic, n = 2, $2 =2f 21 \ μ 3 Third harmonic, n=3, $3 =3f1 Mass per unit length of the string is the ratio of mass to length of the string. m μ=- ‫יד‬ however, m=pV=p Al PAI -=PA where μ=5 A = uniform cross-sectional area of the string P = density of material of the string V = volume of the string m = mass of the string