Physics 3 – Nov 7, 2017 P3 Challenge – What is the critical angle for total internal reflection for a cord of fiber optics used for communications made from glass with an index of refraction of 1.52?

Objectives/Agenda/Assignment 4.5 Standing Waves Agenda: Production of standing waves demo Common Characteristics of standing waves Standing Waves on a string. Standing Waves in an open pipe. Standing Waves in a closed pipe. Standing Waves in a half open pipe Assignment: p189 #34-43

Standing waves demo Demonstration using string vibration to generate standing waves similar to what we saw with the transverse waves on springs. https://www.youtube.com/watch?v=-gr7KmTOrx0 – if demo doesn’t work (also has high speed camera) Relate the wavelength to the length of the string for each standing wave mode to discover the relationship between wavelength and string length. Record as many characteristics / observations of a standing wave as you can. (Compare your answers with the next slide.)

Characteristics of Standing Waves Peaks do not move left or right. There are some points that do not move at all = Nodes. Halfway between nodes, the string will have its maximum amplitude = Antinodes Points within a loop all move in the same direction at a given moment in time. Points in adjacent loops move in the opposite direction at the same moment. The amplitude of oscillations along the string mark out a sine function. A standing wave does not transfer energy.

Interference with reflected wave What’s going on? A single wave interferes with itself after being reflected. It alternates between completely constructively and completely destructively. Only happens at particular frequencies such that the displacement at the boundaries is either zero at all times or in maximum amplitude oscillation.

Standing waves on a string First frequency that forms one loop is called the fundamental frequency because all other frequencies occur at multiples of the fundamental. The fundamental is also called the first harmonic or the n=1 harmonic. F1 = fundamental frequency λ1 = 2L F2 = second harmonic = 2F1 λ2 = L F3 = third harmonic = 3F1 λ3 = 2L/3 Fn = nF1 λn = 2L/n n = mode number

Standing waves on a string The specific frequency measured is dependent on the velocity of the waves on the string. The velocity of waves on a string are given by 𝒗= 𝑻𝒆𝒏𝒔𝒊𝒐𝒏 𝑳𝒊𝒏𝒆𝒂𝒓 𝑫𝒆𝒏𝒔𝒊𝒕𝒚 = 𝑻 𝝁 (not in data packet) Linear density = μ = mass/length 𝒗=𝝀𝒇 𝒗= 𝟐𝑳 𝒏 𝒇 𝒇= 𝒏𝒗 𝟐𝑳 𝜆 𝑛 = 2𝐿 𝑛 A string with linear density of of 0.350 g/m is stretched to 0.65 m with a tension of 78 N. What are the frequencies of the first three harmonics of this string? What are their wavelengths? Sketch each of these harmonics and label the nodes and antinodes.

Waves in pipes Standing waves in pipes are similar to standing waves on strings. There are series of frequencies with corresponding wavelengths. The boundary conditions determine what these standing wave patterns look like. The velocity of waves in pipes is the speed of sound = 343 m/s in air. It is longitudinal wave that interfere with themselves. But they are modeled with sine waves just as the transverse waves on strings were. If a pipe is open at an end, an antinode will be located there. If a pipe is closed at an end, a node will be located there.

Standing waves in an open pipe What are the wavelength and frequency for each of these three harmonics? What are the general formulas for frequency and wavelength? 𝒇 𝒏 = 𝒏𝒗 𝟐𝑳 𝜆 𝑛 = 2𝐿 𝑛 Same as for strings

Standing waves in a closed pipe 𝒇 𝒏 = 𝒏𝒗 𝟐𝑳 𝜆 𝑛 = 2𝐿 𝑛 Same as for strings An open pipe, that is 0.45 m long, what are the frequencies of the first three harmonics of this pipe? What are their wavelengths? Sketch each of these harmonics and label the nodes and antinodes. Do the same for a closed pipe of the same length.

Standing waves is half open pipe Boundary condition requires a node at the closed end and an antinode at the open end. Sketch the fundamental in a half open pipe. Notice the valid modes are still called the first, second, third harmonic etc… But the mode index values can only be ODD values, 1, 3, 5, 7… 𝒇 𝒏 = 𝒏𝒗 𝟒𝑳 𝜆 𝑛 = 4𝐿 𝑛 n = 1,3,5….

Exit slip and homework Exit Slip – Sketch the standing waves for the second harmonic of a wave on a string. Determine the frequency of the second harmonic if the tension in the string is 152 N and the string has a linear density of 2.5 x 10-4 kg/m. L = 0.55 m long string. What’s due? (homework for a homework check next class) p189 #34-43 What’s next? (What to read to prepare for the next class. Start reviewing for the Waves Test 11/21