Question

Learning Goal: To understand standing waves including calculation of lambda and f, and to learn the physical meaning behind some musical terms. The columns in the figure (Figure 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. 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 r wavelength of the nth pattern is denoted lambda_u. The nth pattern has n half-wavelengths along the length of the string, so n lambda_n/2 = L. Thus the wavelength of the nth pattern is lambda_n = 2L/n Part B What is the wavelength of the longest wavelength standing wave pattern that can fit on this guitar string"? Express your answer in centimeters. 1ambda_1 _______ cm

Answer 1

The wavelength of the longest wavelength standing wave pattern that can fit on the guitar string is 120 cm.

Step-by-step explanation:

The wavelength of the longest wavelength standing wave pattern that can fit on the guitar string can be calculated using the formula lambda_n = 2L/n, where L is the length of the string and n is the pattern number. In this case, the length of the string is 60 cm. To find the longest wavelength standing wave pattern, we need to find the pattern number that gives the largest value for lambda_n.

The pattern number for the longest wavelength standing wave pattern is 1. Using the formula, lambda_1 = 2L/1 = 2L = 2 × 60 cm = 120 cm.