APMTH 105 Final Project: Musical Sounds

Strings, drums, and instruments alike are able to produce a number of sounds. We are able to model these sounds by calculating the eigenvalues and eigenfunctions of the Laplacian equation, ∇2u = −λ2u. Through the solutions of this equation, we mathematically modeled the sounds we expected to hear for a string, a square drum, and a circular drum. After examining the resulting sounds and frequencies produced through these calculations, we compared this theoretical data to experimental data obtained from recording similar sounds in the field. We used this information to determine how closely our models resembled the data obtained from real instruments. Finally, we used this data to extrapolate about the sounds produced by a number of other instruments. We were able to then incorporate additional songs into our project by analyzing the presence of the vibrations that we modeled in these environments. 1 Materials and Methods For the well-defined part of the project, we first used our notes and a few online sources to solve the wave equation on a string, square drum, and circular drum. We then tested the predictions made in the solved equation by producing sounds and comparing them to graphs that were made based on the square of the coefficients (power) and lambdas (frequencies). We used an elastic band (to mimic the sound of a string being plucked), a square wooden table (to mimic the sound of a square drum being tapped), and a circular lid (to mimic the sound of a circular drum being tapped). We recorded the sounds on our phones, and then analyzed them using MATLAB. We charted frequency graphs to compare how the differently-shaped drums sound different. For the open-ended part, we found video clips online of a clarinet, piano, trumpet, and violin playing the middle C note. We then analyzed the sound files in the same manner as we did in the well defined part: with MATLAB. We analyzed the frequency plots to determine why different instruments playing the same notes sounded different. After analysis of the four instruments, we found a song online that featured these instruments and analyzed that sound file in MATLAB. By analyzing the frequencies in the song, we can determine which of the four instruments was being played.