Pneumonia is the cause of death for over a million children each year around the world, largely in resource poor regions such as sub-Saharan Africa and remote Asia. One of the biggest challenges faced by pneumonia endemic countries is the absence of a field deployable diagnostic tool that is rapid, low-cost and accurate. In this paper, we address this issue and propose a method to screen pneumonia based on the mathematical analysis of cough sounds. In particular, we propose a novel cough feature inspired by wavelet-based crackle detection work in lung sound analysis. These features are then combined with other mathematical features to develop an automated machine classifier, which can separate pneumonia from a range of other respiratory diseases. Both cough and crackles are symptoms of pneumonia, but their existence alone is not a specific enough marker of the disease. In this paper, we hypothesize that the mathematical analysis of cough sounds allows us to diagnose pneumonia with sufficient sensitivity and specificity. Using a bedside microphone, we collected 815 cough sounds from 91 patients with respiratory illnesses such as pneumonia, asthma, and bronchitis. We extracted wavelet features from cough sounds and combined them with other features such as Mel Cepstral coefficients and non-Gaussianity index. We then trained a logistic regression classifier to separate pneumonia from other diseases. As the reference standard, we used the diagnosis by physicians aided with laboratory and radiological results as deemed necessary for a clinical decision. The methods proposed in this paper achieved a sensitivity and specificity of 94% and 63%, respectively, in separating pneumonia patients from non-pneumonia patients based on wavelet features alone. Combining the wavelets with features from our previous work improves the performance further to 94% and 88% sensitivity and specificity. The performance far surpasses that of the WHO criteria currently in common use in resource-limited settings.