Item 6 Learning Goal: To understand standing waves, including calculation of A and f, and to learn the physical meaning behind some musical terms. The columns in the figure (Eigure 1) show the instantaneous shape of a vibrating guitar string drawn every 1 ms. The guitar string is 60 cm long. The left column shows the guitar string shape as a sinusoidal traveling wave passes through it. Notice that the shape is sinusoidal at all times and specific features, such as the crest indicated with the arrow, travel along the string to the right at a constant speed. The right column shows snapshots of the sinusoidal standing wave formed when this sinusoidal traveling wave passes through an identically shaped wave moving in the opposite direction on the same guitar string. The string is momentarily flat when the underlying traveling waves are exactly out of phase. The shape is sinusoidal with twice the original amplitude when the underlying waves are momentarily in phase. This pattern is called a standing wave because no wave features travel down the length of the string. Figure Time Traveling Wave 0 ms 1 ms 2 ms 3 ms I x= 0cm 118/8 Standing Wave II x= x= 60 cm 0 cm 60 cm 1 of 3 > ▼ Standing waves on a guitar string form when waves traveling down the string reflect off a point where the string is tied down or pressed against the fingerboard. The entire series of distortions may be superimposed on a single figure, like this (Eigure 2), indicating different moments in time using traces of different colors or line styles. Part A What is the wavelength of the standing wave shown on the guitar string? Express your answer in centimeters. ▸ View Available Hint(s) A = 40 cm V Previous Answers Correct Nodes are locations in the standing wave pattern where the string doesn't move at all, and hence the traces on the figure intersect. In between nodes are antinodes, where the string moves up and down the most. This figure(Figure 3) shows the first three standing wave patterns that fit on any string with length L tied down at both ends. A pattern's number n is the number of antinodes it contains. The wavelength of the nth pattern is denoted A. The nth pattern has n half-wavelengths along the length of the string, so n = L Thus the wavelength of the nth pattern is Y Part B A₁ = This standing wave pattern has three antinodes, at x = 10 cm, 30 cm, and 50 cm. The pattern also has four nodes, at z = 0 cm, 20 cm, 40 cm, and 60 cm. Notice that the spacing between adjacent antinodes is only half of one wavelength, not one full wavelength. The same is true for the spacing between adjacent nodes. What is the wavelength of the longest wavelength standing wave pattern that can fit on this guitar string? Express your answer in centimeters. ▸ View Available Hint(s) Submit -VE| ΑΣΦ| < 6 of 15 > 4 WC ? A₁ = 2L cm Review