1. **Problem Statement:** Draw the transmission loss (TL) curve for a double wall construction with the given parameters: gypsum board layers, mineral wool, air cavity, steel studs, sound speed $c=344$ m/s, and air density $\rho_0=1.2$ kg/m$^3$. 2. **Key Concepts and Formula:** Transmission loss (TL) in dB measures how much sound is reduced by a barrier. For a double wall, TL depends on mass, stiffness, damping, and cavity properties. The mass law for a single panel is: $$ TL = 20 \log_{10}(M f) - 47 $$ where $M$ is the surface mass density (kg/m$^2$) and $f$ is frequency (Hz). For double walls with cavity and studs, the TL curve is more complex, involving resonance frequencies and stiffness of the cavity and studs. 3. **Calculate Surface Mass Density $M$ of Gypsum Boards:** Each gypsum board layer thickness $t = 13$ mm = 0.013 m. Density $\rho = 720$ kg/m$^3$. Mass per layer: $$ M_{layer} = \rho \times t = 720 \times 0.013 = 9.36 \text{ kg/m}^2 $$ There are 4 layers (2 on each side): $$ M = 4 \times 9.36 = 37.44 \text{ kg/m}^2 $$ 4. **Air Cavity and Mineral Wool:** Total cavity spacing = 185 mm = 0.185 m. Mineral wool depth = 90 mm = 0.09 m. Air space depth = 95 mm = 0.095 m. 5. **Steel Studs:** 4 studs of cross-section 45 mm × 26 mm. Studs add stiffness and reduce TL at resonance frequencies. 6. **Transmission Loss Curve:** The TL curve is frequency dependent. To sketch it: - Use mass law for low and high frequencies. - Account for resonance dips due to cavity and studs. 7. **Desmos LaTeX Function:** A simplified TL function combining mass law and resonance dips can be approximated as: $$ y = 20 \log_{10}(M f) - 47 - \sum_{i} \frac{A_i}{1 + \left(\frac{f - f_{r,i}}{w_i}\right)^2} $$ where $f_{r,i}$ are resonance frequencies, $A_i$ amplitudes, and $w_i$ widths of dips. Since exact resonance frequencies are not given, the plot will show a general increasing trend with dips. **Final answer:** The transmission loss curve increases approximately as $20 \log_{10}(37.44 f) - 47$ with dips at resonance frequencies due to cavity and studs.