SCI 340 L39 Wave interference Combining Waves Ch. 17

SCI 340 L39 Wave interference Group Work Sketch the wave resulting from the addition of the two waves shown at one instant. 3 –3

SCI 340 L39 Wave interference Group Work Sketch the wave resulting from the addition of the two waves shown at one instant. 3 result –3

SCI 340 L39 Wave interference Constructive: Sum of waves has increased amplitude Destructive: Sum of waves has decreased amplitude Two-wave simulation (not functional) Run at: w1 = 0.2; k1 = 0.2; ampl = 15 integral multiples (half, third, quarter) of lambda: multiply W1, k1 by 2, 3, 4 add same-lambda wave with negative amplitude Beats: slightly vary w2 and k2 together from wave 1 values (0.22 and 0.22; 0.21 and 0.21, etc.) Standing waves (use w of about 0.2; try w = 0.2, k = 0.1)

SCI 340 L39 Wave interference Diffraction § 17.3

SCI 340 L39 Wave interference Objectives Use Huygens’s principle to explain why diffraction occurs. Describe the diffraction of waves around barriers.

SCI 340 L39 Wave interference Diffraction A wave front passing through a barrier bends at the edges Run at: w1 = 0.2; k1 = 0.2; ampl = 15 integral multiples (half, third, quarter) of lambda: multiply w1, k1 by 2, 3, 4 add same-lambda wave with negative amplitude Beats: slightly vary w2 and k2 together from wave 1 values (0.22 and 0.22; 0.21 and 0.21, etc.) Standing waves (use w of about 0.2; try w = 0.2, k = 0.1)

Understanding Diffraction SCI 340 L39 Wave interference Understanding Diffraction Huygens’s Principle: A wave front is composed of “wavelets”

Understanding Diffraction SCI 340 L39 Wave interference Understanding Diffraction Huygens’s Principle: A wave front is composed of “wavelets” that interfere to produce the observed front

Single Slit Diffraction SCI 340 L39 Wave interference Single Slit Diffraction Plane waves encounter a slit in a barrier And diffract as they pass through

SCI 340 L39 Wave interference Diffraction Wavelets interfere to make nodes and antinodes (ripple tank simulation) Run at: w1 = 0.2; k1 = 0.2; ampl = 15 integral multiples (half, third, quarter) of lambda: multiply w1, k1 by 2, 3, 4 add same-lambda wave with negative amplitude Beats: slightly vary w2 and k2 together from wave 1 values (0.22 and 0.22; 0.21 and 0.21, etc.) Standing waves (use w of about 0.2; try w = 0.2, k = 0.1)

SCI 340 L39 Wave interference Where are the Nodes? First minimum (node) at the angle where wave front cancels ½ slit across Half cycle out of phase Destructive interference Run at: w1 = 0.2; k1 = 0.2; ampl = 15 integral multiples (half, third, quarter) of lambda: multiply w1, k1 by 2, 3, 4 add same-lambda wave with negative amplitude Beats: slightly vary w2 and k2 together from wave 1 values (0.22 and 0.22; 0.21 and 0.21, etc.) Standing waves (use w of about 0.2; try w = 0.2, k = 0.1) Reach the detector at the same time D l

First Node Rectangular slit: sin q = l/D Circular aperture: sin q = 1.22 l/D Most energy is between the first nodes A narrow aperture makes a wider dispersion Maximum (sin q) = 1 no nodes if D < l

SCI 340 L39 Wave interference Beats § 17.4

SCI 340 L39 Wave interference Beats Waves of similar frequency combine to give alternating times of constructive and destructive interference Distinctive “waa-waa” sound with beat frequency equal to the difference in frequency of the component waves (Why?) (sound files)

Standing Waves Interference in place § 17.5–17.6

SCI 340 L39 Wave interference Standing Waves Sum of waves of equal amplitude and wavelength traveling in opposite directions Half-wavelength divides exactly into the available space Wave pattern has locations of minimum and maximum variation (nodes and antinodes) (standing longitudinal waves) Run at: w1 = 0.2; k1 = 0.2; ampl = 15 integral multiples (half, third, quarter) of lambda: multiply w1, k1 by 2, 3, 4 add same-lambda wave with negative amplitude Beats: slightly vary w2 and k2 together from wave 1 values (0.22 and 0.22; 0.21 and 0.21, etc.) Standing waves (use w of about 0.2; try w = 0.2, k = 0.1)

SCI 340 L39 Wave interference Resonance Objects have characteristic frequencies at which standing waves are sustained Lowest frequency = fundamental Higher frequencies = overtones Sustained motion is a combination of normal modes

Vibrational Modes: Clamped String SCI 340 L39 Wave interference Vibrational Modes: Clamped String Insert Figure 15.3 from class text Source: Griffith, The Physics of Everyday Phenomena, Figure 15.13

Group Work Add together: a fundamental and second overtone.

SCI 340 L39 Wave interference Group Work Result Add together: a fundamental and first overtone. Approaching a sawtooth wave

SCI 340 L39 Wave interference Group Work Result Add together: a fundamental and second overtone. Approaching a square wave

Combinations of Harmonics SCI 340 L39 Wave interference Combinations of Harmonics Characteristic sounds arise from combining particular harmonics in specific ratios Fourier analysis suimulation flute oboe saxophone

“Closed” and “Open” Tube Modes SCI 340 L39 Wave interference “Closed” and “Open” Tube Modes Source: Halliday, Resnick, and Walker, Fundamentals of Physics, 2003, p 419.

SCI 340 L39 Wave interference Poll Question Which has the lowest fundamental tone? A closed tube of length L. An open tube of length L. Both have the same fundamental tone. Not enough information.

SCI 340 L39 Wave interference Sequence of Harmonics Western musical scale and harmonies are based on overtone series (sound files) Sound files: overtones of open tube or clamped string

Musical Tones Octave higher = 2 the frequency Octave + fifth: 3 the frequency Even-tempered scale = compromise to facilitate transposition 12 (half-) steps per octave Half-step: 2  the frequency 12

Two-Dimensional Waves SCI 340 L39 Wave interference Two-Dimensional Waves Ocean waves Earthquake surface waves Animation rectangular membrane Animation circular membrane