Two signals, one R²

Idea. When the regressors are uncorrelated, the multiple regression splits into separate simple regressions, and the explained variances simply add.

Step 1: standardise. Rescaling variables does not change $R^2$, so assume $S_1$, $S_2$ and $Y$ have mean $0$ and variance $1$. Then $\mathrm{Cov}(S_i,Y)={\rho }_i$ with ${\rho }_1=0.3$, ${\rho }_2=0.4$.

Step 2: coefficients. The normal equations are $\Sigma \beta =c$, where $\Sigma$ is the covariance matrix of the signals and $c$ their covariances with $Y$. Here $\Sigma =I$, so $\beta =c=(0.3,0.4)$.

Step 3: explained variance. $$\begin{array}{rl}\mathrm{Var}(\hat{Y})& =\mathrm{Var}(0.3S_1+0.4S_2)\\ & =0.3^2+0.4^2=0.25,\end{array}$$ so $R^2=0.25=\frac{1}{4}$.

What changes with correlated signals. In general $R^2=c^{\top }{\Sigma }^{-1}c$. If the two signals were positively correlated, part of what they explain would overlap and $R^2$ would be below $0.25$. This is why a new signal is worth more when it is uncorrelated with the ones you already have.